To solve the quadratic equation \( x^2 + 5x + 6 = 0 \), we can factor the quadratic expression.
We are looking for two numbers that multiply to \( 6 \) (the constant term) and add up to \( 5 \) (the coefficient of the linear term). The numbers \( 2 \) and \( 3 \) satisfy these conditions since \( 2 \times 3 = 6 \) and \( 2 + 3 = 5 \).
Thus, we can factor the equation as follows:
\[ (x + 2)(x + 3) = 0 \]
To find the values of \( x \), we set each factor equal to zero:
- \( x + 2 = 0 \) ⟹ \( x = -2 \)
- \( x + 3 = 0 \) ⟹ \( x = -3 \)
Therefore, the solutions to the equation \( x^2 + 5x + 6 = 0 \) are
\[ x = -2 \quad \text{and} \quad x = -3. \]