Solve {l}{(x-1+i)(x^2-ix-(1+i))}{+(-2-2i)}

$$\left. \begin{array} { l } { ( x - 1 + i ) ( x ^ { 2 } - i x - ( 1 + i ) ) } \\ { + ( - 2 - 2 i ) } \end{array} \right.$$

Evaluate

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

Solution Steps

Use the distributive property to multiply x-1+i by x^{2}-ix-\left(1+i\right) and combine like terms.

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

Multiply -1 and 1+i to get -1-i.

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

Use the distributive property to multiply x by x^{2}-ix+\left(-1-i\right).

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

Multiply -1 and 1+i to get -1-i.

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

Use the distributive property to multiply -1+i by x^{2}-ix+\left(-1-i\right).

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

Do the additions.

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

Combine -ix^{2} and \left(-1+i\right)x^{2} to get -x^{2}.

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

Combine \left(-1-i\right)x and \left(1+i\right)x to get 0.

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

Expand

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

Solution Steps

Use the distributive property to multiply x-1+i by x^{2}-ix-\left(1+i\right) and combine like terms.

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

Multiply -1 and 1+i to get -1-i.

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

Use the distributive property to multiply x by x^{2}-ix+\left(-1-i\right).

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

Multiply -1 and 1+i to get -1-i.

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

Use the distributive property to multiply -1+i by x^{2}-ix+\left(-1-i\right).

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

Do the additions.

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

Combine -ix^{2} and \left(-1+i\right)x^{2} to get -x^{2}.

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

Combine \left(-1-i\right)x and \left(1+i\right)x to get 0.

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