Solve xleft(x+2right)^2left(x+1right)^2(x-2)

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

Evaluate

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

Solution Steps

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

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

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

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

Use the distributive property to multiply x by x^{2}+4x+4.

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

Use the distributive property to multiply x^{3}+4x^{2}+4x by x^{2}+2x+1 and combine like terms.

$$\left(x^{5}+6x^{4}+13x^{3}+12x^{2}+4x\right)\left(x-2\right)$$

Use the distributive property to multiply x^{5}+6x^{4}+13x^{3}+12x^{2}+4x by x-2 and combine like terms.

$$x^{6}+4x^{5}+x^{4}-14x^{3}-20x^{2}-8x$$

Expand

$$x^{6}+4x^{5}+x^{4}-14x^{3}-20x^{2}-8x$$

Solution Steps

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

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

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

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

Use the distributive property to multiply x by x^{2}+4x+4.

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

Use the distributive property to multiply x^{3}+4x^{2}+4x by x^{2}+2x+1 and combine like terms.

$$\left(x^{5}+6x^{4}+13x^{3}+12x^{2}+4x\right)\left(x-2\right)$$

Use the distributive property to multiply x^{5}+6x^{4}+13x^{3}+12x^{2}+4x by x-2 and combine like terms.

$$x^{6}+4x^{5}+x^{4}-14x^{3}-20x^{2}-8x$$