To factor the quadratic equation \(x^2 + 5x + 6\), we want to find two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (5).
The two numbers that satisfy these conditions are 2 and 3, because:
- \(2 \times 3 = 6\)
- \(2 + 3 = 5\)
Now, we can write the expression \(x^2 + 5x + 6\) as the product of two binomials:
\[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
So the factored form of the quadratic equation \(x^2 + 5x + 6\) is:
\[ (x + 2)(x + 3) \]