Solve x-8/x(x-6)-2(x-8)*x(x-3)+2(x-6/x+3

$$\frac { x - 8 } { x ( x - 6 ) - 2 ( x - 8 ) } \times \frac { x ( x - 3 ) + 2 ( x - 6 } { x + 3 }$$

Evaluate

$$\frac{\left(x-8\right)\left(x^{3}-7x-12\right)}{\left(x+3\right)\left(x-4\right)^{2}}$$

Short Solution Steps

Use the distributive property to multiply x by x-6.

$$\frac{x-8}{x^{2}-6x-2\left(x-8\right)}\left(x\left(x-3\right)+\frac{2\left(x-6\right)}{x+3}\right)$$

Use the distributive property to multiply -2 by x-8.

$$\frac{x-8}{x^{2}-6x-2x+16}\left(x\left(x-3\right)+\frac{2\left(x-6\right)}{x+3}\right)$$

Combine -6x and -2x to get -8x.

$$\frac{x-8}{x^{2}-8x+16}\left(x\left(x-3\right)+\frac{2\left(x-6\right)}{x+3}\right)$$

Use the distributive property to multiply x by x-3.

$$\frac{x-8}{x^{2}-8x+16}\left(x^{2}-3x+\frac{2\left(x-6\right)}{x+3}\right)$$

Use the distributive property to multiply 2 by x-6.

$$\frac{x-8}{x^{2}-8x+16}\left(x^{2}-3x+\frac{2x-12}{x+3}\right)$$

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

$$\frac{x-8}{x^{2}-8x+16}\left(\frac{\left(x^{2}-3x\right)\left(x+3\right)}{x+3}+\frac{2x-12}{x+3}\right)$$

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

$$\frac{x-8}{x^{2}-8x+16}\times \frac{\left(x^{2}-3x\right)\left(x+3\right)+2x-12}{x+3}$$

Do the multiplications in \left(x^{2}-3x\right)\left(x+3\right)+2x-12.

$$\frac{x-8}{x^{2}-8x+16}\times \frac{x^{3}+3x^{2}-3x^{2}-9x+2x-12}{x+3}$$

Combine like terms in x^{3}+3x^{2}-3x^{2}-9x+2x-12.

$$\frac{x-8}{x^{2}-8x+16}\times \frac{x^{3}-7x-12}{x+3}$$

Multiply \frac{x-8}{x^{2}-8x+16} times \frac{x^{3}-7x-12}{x+3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-8\right)\left(x^{3}-7x-12\right)}{\left(x^{2}-8x+16\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x-8 by each term of x^{3}-7x-12.

$$\frac{x^{4}-7x^{2}-12x-8x^{3}+56x+96}{\left(x^{2}-8x+16\right)\left(x+3\right)}$$

Combine -12x and 56x to get 44x.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{\left(x^{2}-8x+16\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x^{2}-8x+16 by each term of x+3.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{x^{3}+3x^{2}-8x^{2}-24x+16x+48}$$

Combine 3x^{2} and -8x^{2} to get -5x^{2}.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{x^{3}-5x^{2}-24x+16x+48}$$

Combine -24x and 16x to get -8x.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{x^{3}-5x^{2}-8x+48}$$

Expand

$$\frac{x^{4}-8x^{3}-7x^{2}+44x+96}{\left(x+3\right)\left(x^{2}-8x+16\right)}$$

Short Solution Steps

Use the distributive property to multiply x by x-6.

$$\frac{x-8}{x^{2}-6x-2\left(x-8\right)}\left(x\left(x-3\right)+\frac{2\left(x-6\right)}{x+3}\right)$$

Use the distributive property to multiply -2 by x-8.

$$\frac{x-8}{x^{2}-6x-2x+16}\left(x\left(x-3\right)+\frac{2\left(x-6\right)}{x+3}\right)$$

Combine -6x and -2x to get -8x.

$$\frac{x-8}{x^{2}-8x+16}\left(x\left(x-3\right)+\frac{2\left(x-6\right)}{x+3}\right)$$

Use the distributive property to multiply x by x-3.

$$\frac{x-8}{x^{2}-8x+16}\left(x^{2}-3x+\frac{2\left(x-6\right)}{x+3}\right)$$

Use the distributive property to multiply 2 by x-6.

$$\frac{x-8}{x^{2}-8x+16}\left(x^{2}-3x+\frac{2x-12}{x+3}\right)$$

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

$$\frac{x-8}{x^{2}-8x+16}\left(\frac{\left(x^{2}-3x\right)\left(x+3\right)}{x+3}+\frac{2x-12}{x+3}\right)$$

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

$$\frac{x-8}{x^{2}-8x+16}\times \frac{\left(x^{2}-3x\right)\left(x+3\right)+2x-12}{x+3}$$

Do the multiplications in \left(x^{2}-3x\right)\left(x+3\right)+2x-12.

$$\frac{x-8}{x^{2}-8x+16}\times \frac{x^{3}+3x^{2}-3x^{2}-9x+2x-12}{x+3}$$

Combine like terms in x^{3}+3x^{2}-3x^{2}-9x+2x-12.

$$\frac{x-8}{x^{2}-8x+16}\times \frac{x^{3}-7x-12}{x+3}$$

Multiply \frac{x-8}{x^{2}-8x+16} times \frac{x^{3}-7x-12}{x+3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-8\right)\left(x^{3}-7x-12\right)}{\left(x^{2}-8x+16\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x-8 by each term of x^{3}-7x-12.

$$\frac{x^{4}-7x^{2}-12x-8x^{3}+56x+96}{\left(x^{2}-8x+16\right)\left(x+3\right)}$$

Combine -12x and 56x to get 44x.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{\left(x^{2}-8x+16\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x^{2}-8x+16 by each term of x+3.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{x^{3}+3x^{2}-8x^{2}-24x+16x+48}$$

Combine 3x^{2} and -8x^{2} to get -5x^{2}.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{x^{3}-5x^{2}-24x+16x+48}$$

Combine -24x and 16x to get -8x.

$$\frac{x^{4}-7x^{2}+44x-8x^{3}+96}{x^{3}-5x^{2}-8x+48}$$