Solve (x+4)(x-(-1-2i))(x-(-1+2i))

$$( x + 4 ) ( x - ( - 1 - 2 i ) ) ( x - ( - 1 + 2 i ) )$$

Evaluate

$$\left(x+4\right)\left(x+\left(1-2i\right)\right)\left(x+\left(1+2i\right)\right)$$

Solution Steps

The opposite of -1-2i is 1+2i.

$$\left(x+4\right)\left(x+\left(1+2i\right)\right)\left(x-\left(-1+2i\right)\right)$$

Apply the distributive property by multiplying each term of x+4 by each term of x+\left(1+2i\right).

$$\left(x^{2}+\left(1+2i\right)x+4x+\left(4+8i\right)\right)\left(x-\left(-1+2i\right)\right)$$

Combine \left(1+2i\right)x and 4x to get \left(5+2i\right)x.

$$\left(x^{2}+\left(5+2i\right)x+\left(4+8i\right)\right)\left(x-\left(-1+2i\right)\right)$$

Use the distributive property to multiply x^{2}+\left(5+2i\right)x+\left(4+8i\right) by x-\left(-1+2i\right).

$$x^{2}\left(x-\left(-1+2i\right)\right)+\left(5+2i\right)x\left(x-\left(-1+2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Multiply -1 and -1+2i to get 1-2i.

$$x^{2}\left(x+\left(1-2i\right)\right)+\left(5+2i\right)x\left(x-\left(-1+2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

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

$$x^{3}+\left(1-2i\right)x^{2}+\left(5+2i\right)x\left(x-\left(-1+2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Multiply -1 and -1+2i to get 1-2i.

$$x^{3}+\left(1-2i\right)x^{2}+\left(5+2i\right)x\left(x+\left(1-2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Use the distributive property to multiply \left(5+2i\right)x by x+\left(1-2i\right).

$$x^{3}+\left(1-2i\right)x^{2}+\left(5+2i\right)x^{2}+\left(9-8i\right)x+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Combine \left(1-2i\right)x^{2} and \left(5+2i\right)x^{2} to get 6x^{2}.

$$x^{3}+6x^{2}+\left(9-8i\right)x+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Multiply -1 and -1+2i to get 1-2i.

$$x^{3}+6x^{2}+\left(9-8i\right)x+\left(4+8i\right)\left(x+\left(1-2i\right)\right)$$

Use the distributive property to multiply 4+8i by x+\left(1-2i\right).

$$x^{3}+6x^{2}+\left(9-8i\right)x+\left(4+8i\right)x+20$$

Combine \left(9-8i\right)x and \left(4+8i\right)x to get 13x.

$$x^{3}+6x^{2}+13x+20$$

Expand

$$x^{3}+6x^{2}+13x+20$$

Solution Steps

The opposite of -1-2i is 1+2i.

$$\left(x+4\right)\left(x+\left(1+2i\right)\right)\left(x-\left(-1+2i\right)\right)$$

Apply the distributive property by multiplying each term of x+4 by each term of x+\left(1+2i\right).

$$\left(x^{2}+\left(1+2i\right)x+4x+\left(4+8i\right)\right)\left(x-\left(-1+2i\right)\right)$$

Combine \left(1+2i\right)x and 4x to get \left(5+2i\right)x.

$$\left(x^{2}+\left(5+2i\right)x+\left(4+8i\right)\right)\left(x-\left(-1+2i\right)\right)$$

Use the distributive property to multiply x^{2}+\left(5+2i\right)x+\left(4+8i\right) by x-\left(-1+2i\right).

$$x^{2}\left(x-\left(-1+2i\right)\right)+\left(5+2i\right)x\left(x-\left(-1+2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Multiply -1 and -1+2i to get 1-2i.

$$x^{2}\left(x+\left(1-2i\right)\right)+\left(5+2i\right)x\left(x-\left(-1+2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

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

$$x^{3}+\left(1-2i\right)x^{2}+\left(5+2i\right)x\left(x-\left(-1+2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Multiply -1 and -1+2i to get 1-2i.

$$x^{3}+\left(1-2i\right)x^{2}+\left(5+2i\right)x\left(x+\left(1-2i\right)\right)+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Use the distributive property to multiply \left(5+2i\right)x by x+\left(1-2i\right).

$$x^{3}+\left(1-2i\right)x^{2}+\left(5+2i\right)x^{2}+\left(9-8i\right)x+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Combine \left(1-2i\right)x^{2} and \left(5+2i\right)x^{2} to get 6x^{2}.

$$x^{3}+6x^{2}+\left(9-8i\right)x+\left(4+8i\right)\left(x-\left(-1+2i\right)\right)$$

Multiply -1 and -1+2i to get 1-2i.

$$x^{3}+6x^{2}+\left(9-8i\right)x+\left(4+8i\right)\left(x+\left(1-2i\right)\right)$$

Use the distributive property to multiply 4+8i by x+\left(1-2i\right).

$$x^{3}+6x^{2}+\left(9-8i\right)x+\left(4+8i\right)x+20$$

Combine \left(9-8i\right)x and \left(4+8i\right)x to get 13x.

$$x^{3}+6x^{2}+13x+20$$