Solve (x^4-x^2-4x-4)div(x-2)div(x+1)/(4^2-4x-4)div(x-2)div(x+1)

$$\frac { ( x ^ { 4 } - x ^ { 2 } - 4 x - 4 ) \div ( x - 2 ) \div ( x + 1 ) } { ( 4 ^ { 2 } - 4 x - 4 ) \div ( x - 2 ) \div ( x + 1 ) }$$

Evaluate

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

Short Solution Steps

Divide \frac{\frac{x^{4}-x^{2}-4x-4}{x-2}}{x+1} by \frac{\frac{4^{2}-4x-4}{x-2}}{x+1} by multiplying \frac{\frac{x^{4}-x^{2}-4x-4}{x-2}}{x+1} by the reciprocal of \frac{\frac{4^{2}-4x-4}{x-2}}{x+1}.

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

Factor the expressions that are not already factored in \frac{x^{4}-x^{2}-4x-4}{x-2}.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(x^{3}+2x^{2}+3x+2\right)\left(x+1\right)}{\left(x+1\right)\times \frac{16-4x-4}{x-2}}$$

Subtract 4 from 16 to get 12.

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

Express \left(x+1\right)\times \frac{12-4x}{x-2} as a single fraction.

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

Divide \left(x^{3}+2x^{2}+3x+2\right)\left(x+1\right) by \frac{\left(x+1\right)\left(12-4x\right)}{x-2} by multiplying \left(x^{3}+2x^{2}+3x+2\right)\left(x+1\right) by the reciprocal of \frac{\left(x+1\right)\left(12-4x\right)}{x-2}.

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

Cancel out x+1 in both numerator and denominator.

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

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

$$\frac{x^{4}-x^{2}-4x-4}{-4x+12}$$

Expand

$$-\frac{x^{4}-x^{2}-4x-4}{4\left(x-3\right)}$$

Short Solution Steps

Divide \frac{\frac{x^{4}-x^{2}-4x-4}{x-2}}{x+1} by \frac{\frac{4^{2}-4x-4}{x-2}}{x+1} by multiplying \frac{\frac{x^{4}-x^{2}-4x-4}{x-2}}{x+1} by the reciprocal of \frac{\frac{4^{2}-4x-4}{x-2}}{x+1}.

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

Factor the expressions that are not already factored in \frac{x^{4}-x^{2}-4x-4}{x-2}.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(x^{3}+2x^{2}+3x+2\right)\left(x+1\right)}{\left(x+1\right)\times \frac{16-4x-4}{x-2}}$$

Subtract 4 from 16 to get 12.

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

Express \left(x+1\right)\times \frac{12-4x}{x-2} as a single fraction.

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

Divide \left(x^{3}+2x^{2}+3x+2\right)\left(x+1\right) by \frac{\left(x+1\right)\left(12-4x\right)}{x-2} by multiplying \left(x^{3}+2x^{2}+3x+2\right)\left(x+1\right) by the reciprocal of \frac{\left(x+1\right)\left(12-4x\right)}{x-2}.

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

Cancel out x+1 in both numerator and denominator.

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

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

$$\frac{x^{4}-x^{2}-4x-4}{-4x+12}$$