To factorize the expression \(2y^2 - 15xy + 18x^2\), we look for two binomials that, when multiplied, give us the original expression.

The expression can be factored as follows:

\(2y^2 - 15xy + 18x^2\)

First, let's look for two numbers that multiply to give \(2 \cdot 18 = 36\) and add up to give \(-15\). These numbers are \(-3\) and \(-12\).

Now we can write the middle term \(-15xy\) as the sum of \(-3xy\) and \(-12xy\):

\(2y^2 - 3xy - 12xy + 18x^2\)

Next, we group the terms and factor by grouping:

y(2y - 3x) - 6x(2y - 3x)

(2y - 3x)(y - 6x)