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

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

Solve for x

$$x=-1$$

$$x=3$$

Steps Using Factoring

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.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Use the distributive property to multiply 2x+2 by x-1 and combine like terms.

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

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

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

Subtract 2 from 1 to get -1.

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

Subtract 3x^{2} from both sides.

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

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

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

Subtract 2x from both sides.

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

Add 1 to both sides.

$$x^{2}-4-2x+1=0$$

Add -4 and 1 to get -3.

$$x^{2}-3-2x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-2x-3=0$$

To solve the equation, factor x^{2}-2x-3 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-3 b=1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-3\right)\left(x+1\right)$$

To find equation solutions, solve x-3=0 and x+1=0.

$$x=3 x=-1$$

Steps Using Factoring By Grouping

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.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Use the distributive property to multiply 2x+2 by x-1 and combine like terms.

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

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

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

Subtract 2 from 1 to get -1.

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

Subtract 3x^{2} from both sides.

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

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

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

Subtract 2x from both sides.

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

Add 1 to both sides.

$$x^{2}-4-2x+1=0$$

Add -4 and 1 to get -3.

$$x^{2}-3-2x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-2x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-3\right)=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-3 b=1$$

Rewrite x^{2}-2x-3 as \left(x^{2}-3x\right)+\left(x-3\right).

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

Factor out x in x^{2}-3x.

$$x\left(x-3\right)+x-3$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+1\right)$$

To find equation solutions, solve x-3=0 and x+1=0.

$$x=3 x=-1$$

Steps Using the Quadratic Formula

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.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Use the distributive property to multiply 2x+2 by x-1 and combine like terms.

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

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

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

Subtract 2 from 1 to get -1.

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

Subtract 3x^{2} from both sides.

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

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

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

Subtract 2x from both sides.

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

Add 1 to both sides.

$$x^{2}-4-2x+1=0$$

Add -4 and 1 to get -3.

$$x^{2}-3-2x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-2x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-3\right)}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-3\right)}}{2}$$

Multiply -4 times -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+12}}{2}$$

Add 4 to 12.

$$x=\frac{-\left(-2\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-2\right)±4}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±4}{2}$$

Now solve the equation x=\frac{2±4}{2} when ± is plus. Add 2 to 4.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{2±4}{2} when ± is minus. Subtract 4 from 2.

$$x=-\frac{2}{2}$$

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=3 x=-1$$

Steps for Completing the Square

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.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Use the distributive property to multiply 2x+2 by x-1 and combine like terms.

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

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

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

Subtract 2 from 1 to get -1.

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

Subtract 3x^{2} from both sides.

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

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

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

Subtract 2x from both sides.

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

Add 4 to both sides.

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

Add -1 and 4 to get 3.

$$x^{2}-2x=3$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Add 3 to 1.

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

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=2 x-1=-2$$

Add 1 to both sides of the equation.

$$x=3 x=-1$$