To solve the equation \( x^2 + 6x - 12 = 4 \) by factoring, we first need to set the equation equal to zero.
Subtract 4 from both sides:
\[ x^2 + 6x - 12 - 4 = 0 \]
This simplifies to:
\[ x^2 + 6x - 16 = 0 \]
Next, we factor the quadratic expression \( x^2 + 6x - 16 \). We need two numbers that multiply to \(-16\) (the constant term) and add to \(6\) (the coefficient of \(x\)).
The numbers \(8\) and \(-2\) meet these conditions because:
\[ 8 \cdot (-2) = -16 \quad \text{and} \quad 8 + (-2) = 6 \]
Thus, we can factor the quadratic as follows:
\[ (x + 8)(x - 2) = 0 \]
Now, we set each factor equal to zero:
-
\( x + 8 = 0 \)
- This gives \( x = -8 \)
-
\( x - 2 = 0 \)
- This gives \( x = 2 \)
Therefore, the solutions to the equation \( x^2 + 6x - 12 = 4 \) are:
\[ \boxed{-8 \text{ and } 2} \]