To factor the expression \( 45x - 18y + 36 \) completely, we first look for the greatest common factor (GCF) of all the terms.
The coefficients are \( 45, -18, \) and \( 36 \). The GCF of these numbers is \( 9 \).
Now, we can factor \( 9 \) out of the expression:
\[ 45x - 18y + 36 = 9(5x - 2y + 4) \]
Next, let's check if the expression inside the parentheses \( 5x - 2y + 4 \) can be factored further. Since the terms do not have a common factor, and this does not represent a quadratic or any other easily factorable form, we conclude the factoring process here.
Thus, the completely factored form of the expression is:
\[ \boxed{9(5x - 2y + 4)} \]