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

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

Evaluate

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

Solution Steps

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

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

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

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

Use the distributive property to multiply x by x-2.

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

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

$$x^{4}+4x^{3}+4x^{2}-2x^{3}-8x^{2}-8x-1$$

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

$$x^{4}+2x^{3}+4x^{2}-8x^{2}-8x-1$$

Combine 4x^{2} and -8x^{2} to get -4x^{2}.

$$x^{4}+2x^{3}-4x^{2}-8x-1$$

Expand

$$x^{4}+2x^{3}-4x^{2}-8x-1$$

Solution Steps

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

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

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

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

Use the distributive property to multiply x by x-2.

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

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

$$x^{4}+4x^{3}+4x^{2}-2x^{3}-8x^{2}-8x-1$$

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

$$x^{4}+2x^{3}+4x^{2}-8x^{2}-8x-1$$

Combine 4x^{2} and -8x^{2} to get -4x^{2}.

$$x^{4}+2x^{3}-4x^{2}-8x-1$$