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

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

Evaluate

$$8$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x^{2}+2x+1.

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

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

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

Use the distributive property to multiply 2 by x^{2}-2x+1.

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

Use the distributive property to multiply -4 by x+1.

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

Use the distributive property to multiply -4x-4 by x-1 and combine like terms.

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

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

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

Add 2 and 4 to get 6.

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

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

$$4x+6-4x+2$$

Combine 4x and -4x to get 0.

$$6+2$$

Add 6 and 2 to get 8.

$$8$$

Factor

$$2^{3}$$

Solution Steps

Factor out 2.

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

Simplify.

$$8$$