1. The problem is to find the values of $x$ for the quadratic equation $x^2 + 5x + 6 = 0$. 2. We use the factoring method for quadratic equations: if $ax^2 + bx + c = 0$, find two numbers that multiply to $ac$ and add to $b$. 3. Here, $a=1$, $b=5$, and $c=6$. We look for two numbers that multiply to $6$ and add to $5$. These numbers are $2$ and $3$. 4. Factor the quadratic as $(x + 2)(x + 3) = 0$. 5. Set each factor equal to zero: $x + 2 = 0$ or $x + 3 = 0$. 6. Solve for $x$: $x = -2$ or $x = -3$. 7. Therefore, the solutions are $x = -2$ and $x = -3$.