Solve (x-1)^2(x+2)-x(x+1)^2

$$( x - 1 ) ^ { 2 } ( x + 2 ) - x ( x + 1 ) ^ { 2 }$$

Evaluate

$$2-4x-2x^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

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

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

To find the opposite of x^{3}+2x^{2}+x, find the opposite of each term.

$$x^{3}-3x+2-x^{3}-2x^{2}-x$$

Combine x^{3} and -x^{3} to get 0.

$$-3x+2-2x^{2}-x$$

Combine -3x and -x to get -4x.

$$-4x+2-2x^{2}$$

Expand

$$2-4x-2x^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

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

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

To find the opposite of x^{3}+2x^{2}+x, find the opposite of each term.

$$x^{3}-3x+2-x^{3}-2x^{2}-x$$

Combine x^{3} and -x^{3} to get 0.

$$-3x+2-2x^{2}-x$$

Combine -3x and -x to get -4x.

$$-4x+2-2x^{2}$$