Solve x-1/x^2-4*12x-24/x^2-1+2(x+1/x-2):x^2-1/3x-6=frac{3+x}{2x+4}

$$\frac { x - 1 } { x ^ { 2 } - 4 } \cdot \frac { 12 x - 24 } { x ^ { 2 } - 1 } + 2 ( x + \frac { 1 } { x - 2 } ) : \frac { x ^ { 2 } - 1 } { 3 x - 6 } = \frac { 3 + x } { 2 x + 4 }$$

Solve for x

$$x=\frac{3}{11}\approx 0.272727273$$

Solution Steps

Variable x cannot be equal to any of the values -2,-1,1,2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x^{2}-4,x^{2}-1,2x+4.

$$\left(2x^{2}-2\right)\left(x-1\right)\times \frac{12x-24}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express \left(2x^{2}-2\right)\times \frac{12x-24}{x^{2}-1} as a single fraction.

$$\frac{\left(2x^{2}-2\right)\left(12x-24\right)}{x^{2}-1}\left(x-1\right)+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply \frac{\left(2x^{2}-2\right)\left(12x-24\right)}{x^{2}-1} by x-1.

$$\frac{\left(2x^{2}-2\right)\left(12x-24\right)}{x^{2}-1}x-\frac{\left(2x^{2}-2\right)\left(12x-24\right)}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply 2x^{2}-2 by 12x-24.

$$\frac{24x^{3}-48x^{2}-24x+48}{x^{2}-1}x-\frac{\left(2x^{2}-2\right)\left(12x-24\right)}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express \frac{24x^{3}-48x^{2}-24x+48}{x^{2}-1}x as a single fraction.

$$\frac{\left(24x^{3}-48x^{2}-24x+48\right)x}{x^{2}-1}-\frac{\left(2x^{2}-2\right)\left(12x-24\right)}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply 2x^{2}-2 by 12x-24.

$$\frac{\left(24x^{3}-48x^{2}-24x+48\right)x}{x^{2}-1}-\frac{24x^{3}-48x^{2}-24x+48}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Since \frac{\left(24x^{3}-48x^{2}-24x+48\right)x}{x^{2}-1} and \frac{24x^{3}-48x^{2}-24x+48}{x^{2}-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(24x^{3}-48x^{2}-24x+48\right)x-\left(24x^{3}-48x^{2}-24x+48\right)}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Do the multiplications in \left(24x^{3}-48x^{2}-24x+48\right)x-\left(24x^{3}-48x^{2}-24x+48\right).

$$\frac{24x^{4}-48x^{3}-24x^{2}+48x-24x^{3}+48x^{2}+24x-48}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Combine like terms in 24x^{4}-48x^{3}-24x^{2}+48x-24x^{3}+48x^{2}+24x-48.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+6\left(x-2\right)^{2}\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+6\left(x^{2}-4x+4\right)\left(x+2\right)\times 2\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Multiply 6 and 2 to get 12.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+12\left(x^{2}-4x+4\right)\left(x+2\right)\left(x+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-2}{x-2}.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+12\left(x^{2}-4x+4\right)\left(x+2\right)\left(\frac{x\left(x-2\right)}{x-2}+\frac{1}{x-2}\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Since \frac{x\left(x-2\right)}{x-2} and \frac{1}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+12\left(x^{2}-4x+4\right)\left(x+2\right)\times \frac{x\left(x-2\right)+1}{x-2}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Do the multiplications in x\left(x-2\right)+1.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+12\left(x^{2}-4x+4\right)\left(x+2\right)\times \frac{x^{2}-2x+1}{x-2}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express 12\times \frac{x^{2}-2x+1}{x-2} as a single fraction.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\frac{12\left(x^{2}-2x+1\right)}{x-2}\left(x^{2}-4x+4\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply \frac{12\left(x^{2}-2x+1\right)}{x-2} by x^{2}-4x+4.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{12\left(x^{2}-2x+1\right)}{x-2}x^{2}-4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}x+4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply 12 by x^{2}-2x+1.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{12x^{2}-24x+12}{x-2}x^{2}-4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}x+4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express \frac{12x^{2}-24x+12}{x-2}x^{2} as a single fraction.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2}-4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}x+4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply 12 by x^{2}-2x+1.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2}-4\times \frac{12x^{2}-24x+12}{x-2}x+4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express -4\times \frac{12x^{2}-24x+12}{x-2} as a single fraction.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2}+\frac{-4\left(12x^{2}-24x+12\right)}{x-2}x+4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express \frac{-4\left(12x^{2}-24x+12\right)}{x-2}x as a single fraction.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2}+\frac{-4\left(12x^{2}-24x+12\right)x}{x-2}+4\times \frac{12\left(x^{2}-2x+1\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply 12 by x^{2}-2x+1.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2}+\frac{-4\left(12x^{2}-24x+12\right)x}{x-2}+4\times \frac{12x^{2}-24x+12}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express 4\times \frac{12x^{2}-24x+12}{x-2} as a single fraction.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2}+\frac{-4\left(12x^{2}-24x+12\right)x}{x-2}+\frac{4\left(12x^{2}-24x+12\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Since \frac{\left(12x^{2}-24x+12\right)x^{2}}{x-2} and \frac{-4\left(12x^{2}-24x+12\right)x}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{\left(12x^{2}-24x+12\right)x^{2}-4\left(12x^{2}-24x+12\right)x}{x-2}+\frac{4\left(12x^{2}-24x+12\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Do the multiplications in \left(12x^{2}-24x+12\right)x^{2}-4\left(12x^{2}-24x+12\right)x.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{12x^{4}-24x^{3}+12x^{2}-48x^{3}+96x^{2}-48x}{x-2}+\frac{4\left(12x^{2}-24x+12\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Combine like terms in 12x^{4}-24x^{3}+12x^{2}-48x^{3}+96x^{2}-48x.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\left(\frac{12x^{4}-72x^{3}+108x^{2}-48x}{x-2}+\frac{4\left(12x^{2}-24x+12\right)}{x-2}\right)\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Since \frac{12x^{4}-72x^{3}+108x^{2}-48x}{x-2} and \frac{4\left(12x^{2}-24x+12\right)}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\frac{12x^{4}-72x^{3}+108x^{2}-48x+4\left(12x^{2}-24x+12\right)}{x-2}\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Do the multiplications in 12x^{4}-72x^{3}+108x^{2}-48x+4\left(12x^{2}-24x+12\right).

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\frac{12x^{4}-72x^{3}+108x^{2}-48x+48x^{2}-96x+48}{x-2}\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Combine like terms in 12x^{4}-72x^{3}+108x^{2}-48x+48x^{2}-96x+48.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\frac{12x^{4}-72x^{3}+156x^{2}-144x+48}{x-2}\left(x+2\right)=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Express \frac{12x^{4}-72x^{3}+156x^{2}-144x+48}{x-2}\left(x+2\right) as a single fraction.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\frac{\left(12x^{4}-72x^{3}+156x^{2}-144x+48\right)\left(x+2\right)}{x-2}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply 12x^{4}-72x^{3}+156x^{2}-144x+48 by x+2 and combine like terms.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{x^{2}-1}+\frac{12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96}{x-2}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Factor x^{2}-1.

$$\frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{\left(x-1\right)\left(x+1\right)}+\frac{12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96}{x-2}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x-2 is \left(x-2\right)\left(x-1\right)\left(x+1\right). Multiply \frac{24x^{4}-72x^{3}+24x^{2}+72x-48}{\left(x-1\right)\left(x+1\right)} times \frac{x-2}{x-2}. Multiply \frac{12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96}{x-2} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

$$\frac{\left(24x^{4}-72x^{3}+24x^{2}+72x-48\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{\left(12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Since \frac{\left(24x^{4}-72x^{3}+24x^{2}+72x-48\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{\left(12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(24x^{4}-72x^{3}+24x^{2}+72x-48\right)\left(x-2\right)+\left(12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Do the multiplications in \left(24x^{4}-72x^{3}+24x^{2}+72x-48\right)\left(x-2\right)+\left(12x^{5}-48x^{4}+12x^{3}+168x^{2}-240x+96\right)\left(x-1\right)\left(x+1\right).

$$\frac{24x^{5}-48x^{4}-72x^{4}+144x^{3}+24x^{3}-48x^{2}+72x^{2}-144x-48x+96+12x^{7}-12x^{5}-48x^{6}+48x^{4}+12x^{5}-12x^{3}+168x^{4}-168x^{2}-240x^{3}+240x+96x^{2}-96}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Combine like terms in 24x^{5}-48x^{4}-72x^{4}+144x^{3}+24x^{3}-48x^{2}+72x^{2}-144x-48x+96+12x^{7}-12x^{5}-48x^{6}+48x^{4}+12x^{5}-12x^{3}+168x^{4}-168x^{2}-240x^{3}+240x+96x^{2}-96.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=\left(x-2\right)\left(x^{2}-1\right)\left(3+x\right)$$

Use the distributive property to multiply x-2 by x^{2}-1.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=\left(x^{3}-x-2x^{2}+2\right)\left(3+x\right)$$

Use the distributive property to multiply x^{3}-x-2x^{2}+2 by 3+x and combine like terms.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=x^{3}+x^{4}-x-7x^{2}+6$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x^{2}-3x+2\right)\left(x+1\right)}=x^{3}+x^{4}-x-7x^{2}+6$$

Use the distributive property to multiply x^{2}-3x+2 by x+1 and combine like terms.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{x^{3}-2x^{2}-x+2}=x^{3}+x^{4}-x-7x^{2}+6$$

Subtract x^{3} from both sides.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{x^{3}-2x^{2}-x+2}-x^{3}=x^{4}-x-7x^{2}+6$$

Factor x^{3}-2x^{2}-x+2.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-x^{3}=x^{4}-x-7x^{2}+6$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3} times \frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-\frac{x^{3}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=x^{4}-x-7x^{2}+6$$

Since \frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{x^{3}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}-x^{3}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=x^{4}-x-7x^{2}+6$$

Do the multiplications in 24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}-x^{3}\left(x-2\right)\left(x-1\right)\left(x+1\right).

$$\frac{24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}-x^{6}+x^{4}+2x^{5}-2x^{3}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=x^{4}-x-7x^{2}+6$$

Combine like terms in 24x^{5}+96x^{4}-84x^{3}-48x^{2}+48x+12x^{7}-48x^{6}-x^{6}+x^{4}+2x^{5}-2x^{3}.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=x^{4}-x-7x^{2}+6$$

Subtract x^{4} from both sides.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-x^{4}=-x-7x^{2}+6$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{\left(x^{2}-3x+2\right)\left(x+1\right)}-x^{4}=-x-7x^{2}+6$$

Use the distributive property to multiply x^{2}-3x+2 by x+1 and combine like terms.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{x^{3}-2x^{2}-x+2}-x^{4}=-x-7x^{2}+6$$

Factor x^{3}-2x^{2}-x+2.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-x^{4}=-x-7x^{2}+6$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{4} times \frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-\frac{x^{4}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-x-7x^{2}+6$$

Since \frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{x^{4}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}-x^{4}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-x-7x^{2}+6$$

Do the multiplications in 26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}-x^{4}\left(x-2\right)\left(x-1\right)\left(x+1\right).

$$\frac{26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}-x^{7}+x^{5}+2x^{6}-2x^{4}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-x-7x^{2}+6$$

Combine like terms in 26x^{5}+97x^{4}-86x^{3}-48x^{2}-49x^{6}+48x+12x^{7}-x^{7}+x^{5}+2x^{6}-2x^{4}.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-x-7x^{2}+6$$

Add x to both sides.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+x=-7x^{2}+6$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{\left(x^{2}-3x+2\right)\left(x+1\right)}+x=-7x^{2}+6$$

Use the distributive property to multiply x^{2}-3x+2 by x+1 and combine like terms.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{x^{3}-2x^{2}-x+2}+x=-7x^{2}+6$$

Factor x^{3}-2x^{2}-x+2.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+x=-7x^{2}+6$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{x\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-7x^{2}+6$$

Since \frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{x\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}+x\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-7x^{2}+6$$

Do the multiplications in 27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}+x\left(x-2\right)\left(x-1\right)\left(x+1\right).

$$\frac{27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}+x^{4}-x^{2}-2x^{3}+2x}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-7x^{2}+6$$

Combine like terms in 27x^{5}+95x^{4}-86x^{3}-48x^{2}+48x-47x^{6}+11x^{7}+x^{4}-x^{2}-2x^{3}+2x.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=-7x^{2}+6$$

Add 7x^{2} to both sides.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+7x^{2}=6$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{\left(x^{2}-3x+2\right)\left(x+1\right)}+7x^{2}=6$$

Use the distributive property to multiply x^{2}-3x+2 by x+1 and combine like terms.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{x^{3}-2x^{2}-x+2}+7x^{2}=6$$

Factor x^{3}-2x^{2}-x+2.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+7x^{2}=6$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 7x^{2} times \frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}+\frac{7x^{2}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=6$$

Since \frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{7x^{2}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}+7x^{2}\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=6$$

Do the multiplications in 27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}+7x^{2}\left(x-2\right)\left(x-1\right)\left(x+1\right).

$$\frac{27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}+7x^{5}-7x^{3}-14x^{4}+14x^{2}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=6$$

Combine like terms in 27x^{5}+96x^{4}-88x^{3}-49x^{2}-47x^{6}+50x+11x^{7}+7x^{5}-7x^{3}-14x^{4}+14x^{2}.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=6$$

Subtract 6 from both sides.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-6=0$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{\left(x^{2}-3x+2\right)\left(x+1\right)}-6=0$$

Use the distributive property to multiply x^{2}-3x+2 by x+1 and combine like terms.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{x^{3}-2x^{2}-x+2}-6=0$$

Factor x^{3}-2x^{2}-x+2.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-6=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6 times \frac{\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}-\frac{6\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=0$$

Since \frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} and \frac{6\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}-6\left(x-2\right)\left(x-1\right)\left(x+1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=0$$

Do the multiplications in 34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}-6\left(x-2\right)\left(x-1\right)\left(x+1\right).

$$\frac{34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}-6x^{3}+6x+12x^{2}-12}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=0$$

Combine like terms in 34x^{5}+82x^{4}-95x^{3}-35x^{2}+50x-47x^{6}+11x^{7}-6x^{3}+6x+12x^{2}-12.

$$\frac{34x^{5}+82x^{4}-101x^{3}-23x^{2}-47x^{6}+56x+11x^{7}-12}{\left(x-2\right)\left(x-1\right)\left(x+1\right)}=0$$

Variable x cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right).

$$34x^{5}+82x^{4}-101x^{3}-23x^{2}-47x^{6}+56x+11x^{7}-12=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$11x^{7}-47x^{6}+34x^{5}+82x^{4}-101x^{3}-23x^{2}+56x-12=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -12 and q divides the leading coefficient 11. List all candidates \frac{p}{q}.

$$±\frac{12}{11},±12,±\frac{6}{11},±6,±\frac{4}{11},±4,±\frac{3}{11},±3,±\frac{2}{11},±2,±\frac{1}{11},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 11x^{7}-47x^{6}+34x^{5}+82x^{4}-101x^{3}-23x^{2}+56x-12 by x-1 to get 11x^{6}-36x^{5}-2x^{4}+80x^{3}-21x^{2}-44x+12. Solve the equation where the result equals to 0.

$$11x^{6}-36x^{5}-2x^{4}+80x^{3}-21x^{2}-44x+12=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 12 and q divides the leading coefficient 11. List all candidates \frac{p}{q}.

$$±\frac{12}{11},±12,±\frac{6}{11},±6,±\frac{4}{11},±4,±\frac{3}{11},±3,±\frac{2}{11},±2,±\frac{1}{11},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 11x^{6}-36x^{5}-2x^{4}+80x^{3}-21x^{2}-44x+12 by x-1 to get 11x^{5}-25x^{4}-27x^{3}+53x^{2}+32x-12. Solve the equation where the result equals to 0.

$$11x^{5}-25x^{4}-27x^{3}+53x^{2}+32x-12=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -12 and q divides the leading coefficient 11. List all candidates \frac{p}{q}.

$$±\frac{12}{11},±12,±\frac{6}{11},±6,±\frac{4}{11},±4,±\frac{3}{11},±3,±\frac{2}{11},±2,±\frac{1}{11},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 11x^{5}-25x^{4}-27x^{3}+53x^{2}+32x-12 by x+1 to get 11x^{4}-36x^{3}+9x^{2}+44x-12. Solve the equation where the result equals to 0.

$$11x^{4}-36x^{3}+9x^{2}+44x-12=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -12 and q divides the leading coefficient 11. List all candidates \frac{p}{q}.

$$±\frac{12}{11},±12,±\frac{6}{11},±6,±\frac{4}{11},±4,±\frac{3}{11},±3,±\frac{2}{11},±2,±\frac{1}{11},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 11x^{4}-36x^{3}+9x^{2}+44x-12 by x+1 to get 11x^{3}-47x^{2}+56x-12. Solve the equation where the result equals to 0.

$$11x^{3}-47x^{2}+56x-12=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -12 and q divides the leading coefficient 11. List all candidates \frac{p}{q}.

$$±\frac{12}{11},±12,±\frac{6}{11},±6,±\frac{4}{11},±4,±\frac{3}{11},±3,±\frac{2}{11},±2,±\frac{1}{11},±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 11x^{3}-47x^{2}+56x-12 by x-2 to get 11x^{2}-25x+6. Solve the equation where the result equals to 0.

$$11x^{2}-25x+6=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 11 for a, -25 for b, and 6 for c in the quadratic formula.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 11\times 6}}{2\times 11}$$

Do the calculations.

$$x=\frac{25±19}{22}$$

Solve the equation 11x^{2}-25x+6=0 when ± is plus and when ± is minus.

$$x=\frac{3}{11} x=2$$

Remove the values that the variable cannot be equal to.

$$x=\frac{3}{11}$$

List all found solutions.

$$x=1 x=-1 x=2 x=\frac{3}{11}$$

Variable x cannot be equal to any of the values 1,-1,2.

$$x=\frac{3}{11}$$