To factor the quadratic expression \( z^2 + 5z + 4 \), we need to find two numbers that multiply to the constant term (4) and add up to the coefficient of the linear term (5).
The numbers that work here are 4 and 1, since:
- \( 4 \times 1 = 4 \)
- \( 4 + 1 = 5 \)
Therefore, we can factor the expression as:
\[ (z + 4)(z + 1) \]
So, the factored form of \( z^2 + 5z + 4 \) is \( (z + 4)(z + 1) \).