Solve 2-x/x^2+xdivx^2-3x/x^2+2x+1*9-x/4-2

$$\frac { 2 - x } { x ^ { 2 } + x } \div \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 2 x + 1 } \cdot \frac { 9 - x } { 4 - 2 }$$

Evaluate

$$\frac{\left(2-x\right)\left(9-x\right)\left(x+1\right)}{2\left(x-3\right)x^{2}}$$

Short Solution Steps

Divide \frac{2-x}{x^{2}+x} by \frac{x^{2}-3x}{x^{2}+2x+1} by multiplying \frac{2-x}{x^{2}+x} by the reciprocal of \frac{x^{2}-3x}{x^{2}+2x+1}.

$$\frac{\left(2-x\right)\left(x^{2}+2x+1\right)}{\left(x^{2}+x\right)\left(x^{2}-3x\right)}\times \frac{9-x}{4-2}$$

Factor the expressions that are not already factored in \frac{\left(2-x\right)\left(x^{2}+2x+1\right)}{\left(x^{2}+x\right)\left(x^{2}-3x\right)}.

$$\frac{\left(-x+2\right)\left(x+1\right)^{2}}{\left(x-3\right)\left(x+1\right)x^{2}}\times \frac{9-x}{4-2}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{\left(x+1\right)\left(-x+2\right)}{\left(x-3\right)x^{2}}\times \frac{9-x}{4-2}$$

Subtract 2 from 4 to get 2.

$$\frac{\left(x+1\right)\left(-x+2\right)}{\left(x-3\right)x^{2}}\times \frac{9-x}{2}$$

Multiply \frac{\left(x+1\right)\left(-x+2\right)}{\left(x-3\right)x^{2}} times \frac{9-x}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+1\right)\left(-x+2\right)\left(9-x\right)}{\left(x-3\right)x^{2}\times 2}$$

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

$$\frac{\left(-x^{2}+x+2\right)\left(9-x\right)}{\left(x-3\right)x^{2}\times 2}$$

Use the distributive property to multiply -x^{2}+x+2 by 9-x and combine like terms.

$$\frac{-10x^{2}+x^{3}+7x+18}{\left(x-3\right)x^{2}\times 2}$$

Use the distributive property to multiply x-3 by x^{2}.

$$\frac{-10x^{2}+x^{3}+7x+18}{\left(x^{3}-3x^{2}\right)\times 2}$$

Use the distributive property to multiply x^{3}-3x^{2} by 2.

$$\frac{-10x^{2}+x^{3}+7x+18}{2x^{3}-6x^{2}}$$

Expand

$$\frac{x^{3}-10x^{2}+7x+18}{2x\left(x^{2}-3x\right)}$$

Short Solution Steps

Divide \frac{2-x}{x^{2}+x} by \frac{x^{2}-3x}{x^{2}+2x+1} by multiplying \frac{2-x}{x^{2}+x} by the reciprocal of \frac{x^{2}-3x}{x^{2}+2x+1}.

$$\frac{\left(2-x\right)\left(x^{2}+2x+1\right)}{\left(x^{2}+x\right)\left(x^{2}-3x\right)}\times \frac{9-x}{4-2}$$

Factor the expressions that are not already factored in \frac{\left(2-x\right)\left(x^{2}+2x+1\right)}{\left(x^{2}+x\right)\left(x^{2}-3x\right)}.

$$\frac{\left(-x+2\right)\left(x+1\right)^{2}}{\left(x-3\right)\left(x+1\right)x^{2}}\times \frac{9-x}{4-2}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{\left(x+1\right)\left(-x+2\right)}{\left(x-3\right)x^{2}}\times \frac{9-x}{4-2}$$

Subtract 2 from 4 to get 2.

$$\frac{\left(x+1\right)\left(-x+2\right)}{\left(x-3\right)x^{2}}\times \frac{9-x}{2}$$

Multiply \frac{\left(x+1\right)\left(-x+2\right)}{\left(x-3\right)x^{2}} times \frac{9-x}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x+1\right)\left(-x+2\right)\left(9-x\right)}{\left(x-3\right)x^{2}\times 2}$$

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

$$\frac{\left(-x^{2}+x+2\right)\left(9-x\right)}{\left(x-3\right)x^{2}\times 2}$$

Use the distributive property to multiply -x^{2}+x+2 by 9-x and combine like terms.

$$\frac{-10x^{2}+x^{3}+7x+18}{\left(x-3\right)x^{2}\times 2}$$

Use the distributive property to multiply x-3 by x^{2}.

$$\frac{-10x^{2}+x^{3}+7x+18}{\left(x^{3}-3x^{2}\right)\times 2}$$

Use the distributive property to multiply x^{3}-3x^{2} by 2.

$$\frac{-10x^{2}+x^{3}+7x+18}{2x^{3}-6x^{2}}$$