Solve x2 + 5x + 6 = 0.

Step-by-step solution

1. 1

   Set up the factoring search. For a monic quadratic $x^2+bx+c$ we need two numbers whose product is $c=6$ and whose sum is $b=5$. Because $c>0$ and $b>0$, both numbers must be positive.

2. 2

   Find the pair. The positive factor pairs of $6$ are $(1,6)$ summing to $7$, and $(2,3)$ summing to $5$. The second pair works:

   $2\cdot 3=6,\qquad 2+3=5$

3. 3

   Write the factorization.

   $x^2+5x+6=(x+2)(x+3)=0$

4. 4

   Apply the zero product property. A product of real numbers is zero only if one of the factors is zero:

   $x+2=0\quad\text{or}\quad x+3=0\quad\Rightarrow\quad x=-2\ \text{or}\ x=-3$

   Both branches must be kept — the equation has two distinct roots.

5. 5

   Check both, and confirm with Vieta. $(-2)^2+5(-2)+6=4-10+6=0$ ✓ and $(-3)^2+5(-3)+6=9-15+6=0$ ✓. The roots also sum to $-5=-b$ and multiply to $6=c$, as they must.