Solve (1+x)(1+x)(x-1)(x-1)

$$( 1 + x ) ( 1 + x ) ( x - 1 ) ( x - 1 )$$

Evaluate

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

Solution Steps

Multiply 1+x and 1+x to get \left(1+x\right)^{2}.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

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

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

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

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

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

$$x^{2}-2x+1+2x^{3}-4x^{2}+2x+x^{4}-2x^{3}+x^{2}$$

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

$$-3x^{2}-2x+1+2x^{3}+2x+x^{4}-2x^{3}+x^{2}$$

Combine -2x and 2x to get 0.

$$-3x^{2}+1+2x^{3}+x^{4}-2x^{3}+x^{2}$$

Combine 2x^{3} and -2x^{3} to get 0.

$$-3x^{2}+1+x^{4}+x^{2}$$

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

$$-2x^{2}+1+x^{4}$$

Expand

$$x^{4}-2x^{2}+1$$

Solution Steps

Multiply 1+x and 1+x to get \left(1+x\right)^{2}.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

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

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

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

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

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

$$x^{2}-2x+1+2x^{3}-4x^{2}+2x+x^{4}-2x^{3}+x^{2}$$

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

$$-3x^{2}-2x+1+2x^{3}+2x+x^{4}-2x^{3}+x^{2}$$

Combine -2x and 2x to get 0.

$$-3x^{2}+1+2x^{3}+x^{4}-2x^{3}+x^{2}$$

Combine 2x^{3} and -2x^{3} to get 0.

$$-3x^{2}+1+x^{4}+x^{2}$$

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

$$-2x^{2}+1+x^{4}$$