To factor the expression \( 45x - 18y + 36 \) completely, we can start by looking for common factors in the terms.
First, let's rewrite the expression:
\[ 45x - 18y + 36 \]
Next, we can factor out the greatest common factor (GCF) of the coefficients \(45\), \(-18\), and \(36\). The GCF is \(9\):
\[ 9(5x - 2y + 4) \]
At this point, we have factored out \(9\), and now we need to see if the expression inside the parentheses, \(5x - 2y + 4\), can be factored further.
However, \(5x - 2y + 4\) does not factor nicely over the integers or using simple factoring techniques. Therefore, the expression is completely factored as:
\[ \boxed{9(5x - 2y + 4)} \]