Question

solve for all values of x by factoring.
\\( x^2 + 2x = -2x \\)

Explanation:

Step1: Move all terms to left side

First, we need to get all terms on one side of the equation to set it equal to zero. Add \(2x\) to both sides of the equation \(x^{2}+2x = - 2x\).
\(x^{2}+2x + 2x=0\)
Simplify the left - hand side: \(x^{2}+4x = 0\)

Step2: Factor the left - hand side

We can factor out the greatest common factor (GCF) from the terms \(x^{2}\) and \(4x\). The GCF of \(x^{2}\) and \(4x\) is \(x\).
Using the distributive property \(ab+ac=a(b + c)\) (in reverse), we have \(x(x + 4)=0\)

Step3: Apply the zero - product property

The zero - product property states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\) (or both).
For \(x(x + 4)=0\), we set each factor equal to zero:

• Case 1: \(x=0\)
• Case 2: \(x + 4=0\)

Solve the second equation: Subtract 4 from both sides, we get \(x=-4\)

Answer:

\(x = 0\) or \(x=-4\)