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

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

Evaluate

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

Solution Steps

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

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

Square x^{2}+x+1.

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

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

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

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

$$x^{7}+2x^{6}-2x^{4}-4x^{3}+x+2$$

Expand

$$x^{7}+2x^{6}-2x^{4}-4x^{3}+x+2$$

Solution Steps

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

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

Square x^{2}+x+1.

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

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

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

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

$$x^{7}+2x^{6}-2x^{4}-4x^{3}+x+2$$