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

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

Evaluate

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

Solution Steps

Apply the distributive property by multiplying each term of x-1-i by each term of x-1+i.

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

Combine the real and imaginary parts in 1-i+i+1.

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

Do the additions in 1+1+\left(-1+1\right)i.

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

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

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

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

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

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

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

Apply the distributive property by multiplying each term of x^{2}-2x+2 by each term of x-2.

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

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

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

Combine 4x and 2x to get 6x.

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

Apply the distributive property by multiplying each term of x^{3}-4x^{2}+6x-4 by each term of x+1.

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

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

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

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

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

Combine 6x and -4x to get 2x.

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

Expand

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

Solution Steps

Apply the distributive property by multiplying each term of x-1-i by each term of x-1+i.

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

Combine the real and imaginary parts in 1-i+i+1.

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

Do the additions in 1+1+\left(-1+1\right)i.

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

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

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

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

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

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

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

Apply the distributive property by multiplying each term of x^{2}-2x+2 by each term of x-2.

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

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

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

Combine 4x and 2x to get 6x.

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

Apply the distributive property by multiplying each term of x^{3}-4x^{2}+6x-4 by each term of x+1.

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

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

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

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

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

Combine 6x and -4x to get 2x.

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