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

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

Evaluate

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

Solution Steps

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

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

Consider \left(2x+2\right)\left(2x-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}+2x+1-4-4x\left(x-1\right)$$

Subtract 4 from 1 to get -3.

$$5x^{2}+2x-3-4x\left(x-1\right)$$

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

$$5x^{2}+2x-3-4x^{2}+4x$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

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

Combine 2x and 4x to get 6x.

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

Expand

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

Solution Steps

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

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

Consider \left(2x+2\right)\left(2x-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}+2x+1-4-4x\left(x-1\right)$$

Subtract 4 from 1 to get -3.

$$5x^{2}+2x-3-4x\left(x-1\right)$$

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

$$5x^{2}+2x-3-4x^{2}+4x$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

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

Combine 2x and 4x to get 6x.

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