Solve 7x-1/x^2-2x-3-6x+1/x^2-x-26

$$\frac{ 7x-1 }{ { x }^{ 2 } -2x-3 } - \frac{ 6x+1 }{ { x }^{ 2 } -x-2 } 6$$

Evaluate

$$-\frac{29x^{2}-87x-20}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Short Solution Steps

Express \frac{6x+1}{x^{2}-x-2}\times 6 as a single fraction.

$$\frac{7x-1}{x^{2}-2x-3}-\frac{\left(6x+1\right)\times 6}{x^{2}-x-2}$$

Use the distributive property to multiply 6x+1 by 6.

$$\frac{7x-1}{x^{2}-2x-3}-\frac{36x+6}{x^{2}-x-2}$$

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

$$\frac{7x-1}{\left(x-3\right)\left(x+1\right)}-\frac{36x+6}{\left(x-2\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+1\right) and \left(x-2\right)\left(x+1\right) is \left(x-3\right)\left(x-2\right)\left(x+1\right). Multiply \frac{7x-1}{\left(x-3\right)\left(x+1\right)} times \frac{x-2}{x-2}. Multiply \frac{36x+6}{\left(x-2\right)\left(x+1\right)} times \frac{x-3}{x-3}.

$$\frac{\left(7x-1\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}-\frac{\left(36x+6\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Since \frac{\left(7x-1\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)} and \frac{\left(36x+6\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(7x-1\right)\left(x-2\right)-\left(36x+6\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Do the multiplications in \left(7x-1\right)\left(x-2\right)-\left(36x+6\right)\left(x-3\right).

$$\frac{7x^{2}-14x-x+2-36x^{2}+108x-6x+18}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Combine like terms in 7x^{2}-14x-x+2-36x^{2}+108x-6x+18.

$$\frac{-29x^{2}+87x+20}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Expand \left(x-3\right)\left(x-2\right)\left(x+1\right).

$$\frac{-29x^{2}+87x+20}{x^{3}-4x^{2}+x+6}$$

Expand

$$\frac{20+87x-29x^{2}}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Short Solution Steps

Express \frac{6x+1}{x^{2}-x-2}\times 6 as a single fraction.

$$\frac{7x-1}{x^{2}-2x-3}-\frac{\left(6x+1\right)\times 6}{x^{2}-x-2}$$

Use the distributive property to multiply 6x+1 by 6.

$$\frac{7x-1}{x^{2}-2x-3}-\frac{36x+6}{x^{2}-x-2}$$

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

$$\frac{7x-1}{\left(x-3\right)\left(x+1\right)}-\frac{36x+6}{\left(x-2\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+1\right) and \left(x-2\right)\left(x+1\right) is \left(x-3\right)\left(x-2\right)\left(x+1\right). Multiply \frac{7x-1}{\left(x-3\right)\left(x+1\right)} times \frac{x-2}{x-2}. Multiply \frac{36x+6}{\left(x-2\right)\left(x+1\right)} times \frac{x-3}{x-3}.

$$\frac{\left(7x-1\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}-\frac{\left(36x+6\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Since \frac{\left(7x-1\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)} and \frac{\left(36x+6\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(7x-1\right)\left(x-2\right)-\left(36x+6\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Do the multiplications in \left(7x-1\right)\left(x-2\right)-\left(36x+6\right)\left(x-3\right).

$$\frac{7x^{2}-14x-x+2-36x^{2}+108x-6x+18}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Combine like terms in 7x^{2}-14x-x+2-36x^{2}+108x-6x+18.

$$\frac{-29x^{2}+87x+20}{\left(x-3\right)\left(x-2\right)\left(x+1\right)}$$

Expand \left(x-3\right)\left(x-2\right)\left(x+1\right).

$$\frac{-29x^{2}+87x+20}{x^{3}-4x^{2}+x+6}$$