To factor the trinomial \( 5x^2 + 33x + 18 \), we look for two numbers that multiply to \( 5 \times 18 = 90 \) and add to \( 33 \).
The two numbers that work are \( 30 \) and \( 3 \) since:
- \( 30 + 3 = 33 \)
- \( 30 \times 3 = 90 \)
Next, we rewrite the middle term using these two numbers:
\[ 5x^2 + 30x + 3x + 18 \]
Now, we can group the terms:
\[ (5x^2 + 30x) + (3x + 18) \]
Factoring each group gives:
\[ 5x(x + 6) + 3(x + 6) \]
Now we can factor out the common term \( (x + 6) \):
\[ (5x + 3)(x + 6) \]
Thus, the factored form of the trinomial \( 5x^2 + 33x + 18 \) is:
\[ \boxed{(5x + 3)(x + 6)} \]