Solve (x+4)(x-2)(x-(3-i))(x-(3+i))

$$( x + 4 ) ( x - 2 ) ( x - ( 3 - i ) ) ( x - ( 3 + i ) )$$

Evaluate

$$\left(x-2\right)\left(x+4\right)\left(x+\left(-3-i\right)\right)\left(x+\left(-3+i\right)\right)$$

Solution Steps

Apply the distributive property by multiplying each term of x+4 by each term of x-2.

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

Combine -2x and 4x to get 2x.

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

Use the distributive property to multiply x^{2}+2x-8 by x-\left(3-i\right).

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

Use the distributive property to multiply x^{2}\left(x-\left(3-i\right)\right)+2x\left(x-\left(3-i\right)\right)-8\left(x-\left(3-i\right)\right) by x-\left(3+i\right).

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

Multiply -1 and 3-i to get -3+i.

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

Multiply -1 and 3+i to get -3-i.

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

Use the distributive property to multiply x^{2} by x+\left(-3+i\right).

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

Apply the distributive property by multiplying each term of x^{3}+\left(-3+i\right)x^{2} by each term of x+\left(-3-i\right).

$$x^{4}+\left(-3-i\right)x^{3}+\left(-3+i\right)x^{3}+10x^{2}+2x\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine \left(-3-i\right)x^{3} and \left(-3+i\right)x^{3} to get -6x^{3}.

$$x^{4}-6x^{3}+10x^{2}+2x\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Multiply -1 and 3-i to get -3+i.

$$x^{4}-6x^{3}+10x^{2}+2x\left(x+\left(-3+i\right)\right)\left(x-\left(3+i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Multiply -1 and 3+i to get -3-i.

$$x^{4}-6x^{3}+10x^{2}+2x\left(x+\left(-3+i\right)\right)\left(x+\left(-3-i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Use the distributive property to multiply 2x by x+\left(-3+i\right).

$$x^{4}-6x^{3}+10x^{2}+\left(2x^{2}+\left(-6+2i\right)x\right)\left(x+\left(-3-i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Apply the distributive property by multiplying each term of 2x^{2}+\left(-6+2i\right)x by each term of x+\left(-3-i\right).

$$x^{4}-6x^{3}+10x^{2}+2x^{3}+\left(-6-2i\right)x^{2}+\left(-6+2i\right)x^{2}+20x-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine \left(-6-2i\right)x^{2} and \left(-6+2i\right)x^{2} to get -12x^{2}.

$$x^{4}-6x^{3}+10x^{2}+2x^{3}-12x^{2}+20x-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine -6x^{3} and 2x^{3} to get -4x^{3}.

$$x^{4}-4x^{3}+10x^{2}-12x^{2}+20x-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine 10x^{2} and -12x^{2} to get -2x^{2}.

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

Multiply -1 and 3-i to get -3+i.

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

Multiply -1 and 3+i to get -3-i.

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

Use the distributive property to multiply -8 by x+\left(-3+i\right).

$$x^{4}-4x^{3}-2x^{2}+20x+\left(-8x+\left(24-8i\right)\right)\left(x+\left(-3-i\right)\right)$$

Apply the distributive property by multiplying each term of -8x+\left(24-8i\right) by each term of x+\left(-3-i\right).

$$x^{4}-4x^{3}-2x^{2}+20x-8x^{2}+\left(24+8i\right)x+\left(24-8i\right)x-80$$

Combine \left(24+8i\right)x and \left(24-8i\right)x to get 48x.

$$x^{4}-4x^{3}-2x^{2}+20x-8x^{2}+48x-80$$

Combine -2x^{2} and -8x^{2} to get -10x^{2}.

$$x^{4}-4x^{3}-10x^{2}+20x+48x-80$$

Combine 20x and 48x to get 68x.

$$x^{4}-4x^{3}-10x^{2}+68x-80$$

Expand

$$x^{4}-4x^{3}-10x^{2}+68x-80$$

Solution Steps

Apply the distributive property by multiplying each term of x+4 by each term of x-2.

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

Combine -2x and 4x to get 2x.

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

Use the distributive property to multiply x^{2}+2x-8 by x-\left(3-i\right).

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

Use the distributive property to multiply x^{2}\left(x-\left(3-i\right)\right)+2x\left(x-\left(3-i\right)\right)-8\left(x-\left(3-i\right)\right) by x-\left(3+i\right).

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

Multiply -1 and 3-i to get -3+i.

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

Multiply -1 and 3+i to get -3-i.

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

Use the distributive property to multiply x^{2} by x+\left(-3+i\right).

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

Apply the distributive property by multiplying each term of x^{3}+\left(-3+i\right)x^{2} by each term of x+\left(-3-i\right).

$$x^{4}+\left(-3-i\right)x^{3}+\left(-3+i\right)x^{3}+10x^{2}+2x\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine \left(-3-i\right)x^{3} and \left(-3+i\right)x^{3} to get -6x^{3}.

$$x^{4}-6x^{3}+10x^{2}+2x\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Multiply -1 and 3-i to get -3+i.

$$x^{4}-6x^{3}+10x^{2}+2x\left(x+\left(-3+i\right)\right)\left(x-\left(3+i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Multiply -1 and 3+i to get -3-i.

$$x^{4}-6x^{3}+10x^{2}+2x\left(x+\left(-3+i\right)\right)\left(x+\left(-3-i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Use the distributive property to multiply 2x by x+\left(-3+i\right).

$$x^{4}-6x^{3}+10x^{2}+\left(2x^{2}+\left(-6+2i\right)x\right)\left(x+\left(-3-i\right)\right)-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Apply the distributive property by multiplying each term of 2x^{2}+\left(-6+2i\right)x by each term of x+\left(-3-i\right).

$$x^{4}-6x^{3}+10x^{2}+2x^{3}+\left(-6-2i\right)x^{2}+\left(-6+2i\right)x^{2}+20x-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine \left(-6-2i\right)x^{2} and \left(-6+2i\right)x^{2} to get -12x^{2}.

$$x^{4}-6x^{3}+10x^{2}+2x^{3}-12x^{2}+20x-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine -6x^{3} and 2x^{3} to get -4x^{3}.

$$x^{4}-4x^{3}+10x^{2}-12x^{2}+20x-8\left(x-\left(3-i\right)\right)\left(x-\left(3+i\right)\right)$$

Combine 10x^{2} and -12x^{2} to get -2x^{2}.

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

Multiply -1 and 3-i to get -3+i.

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

Multiply -1 and 3+i to get -3-i.

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

Use the distributive property to multiply -8 by x+\left(-3+i\right).

$$x^{4}-4x^{3}-2x^{2}+20x+\left(-8x+\left(24-8i\right)\right)\left(x+\left(-3-i\right)\right)$$

Apply the distributive property by multiplying each term of -8x+\left(24-8i\right) by each term of x+\left(-3-i\right).

$$x^{4}-4x^{3}-2x^{2}+20x-8x^{2}+\left(24+8i\right)x+\left(24-8i\right)x-80$$

Combine \left(24+8i\right)x and \left(24-8i\right)x to get 48x.

$$x^{4}-4x^{3}-2x^{2}+20x-8x^{2}+48x-80$$

Combine -2x^{2} and -8x^{2} to get -10x^{2}.

$$x^{4}-4x^{3}-10x^{2}+20x+48x-80$$

Combine 20x and 48x to get 68x.

$$x^{4}-4x^{3}-10x^{2}+68x-80$$