Identify the quadratic expression.

The expression to factorize is $z^2 + 5z + 6$.
We need to find two numbers that multiply to the constant term (6) and add up to the coefficient of the middle term (5).

Let the two numbers be $a$ and $b$.
We need $a \times b = 6$ and $a + b = 5$.
The pairs of factors for 6 are (1, 6), (2, 3), (-1, -6), (-2, -3).
Checking their sums:
$1 + 6 = 7$
$2 + 3 = 5$
$-1 + (-6) = -7$
$-2 + (-3) = -5$
The numbers that satisfy both conditions are 2 and 3.

Group the terms and factor out the common factors.