After finding $x$, substitute it into the original expression. Evaluate the repeated binomials first rather than expanding the entire polynomial.

The value from the original expression equals a fraction whose numerator is $a+b+c$ and whose denominator is $18$.

Algebra is faster here because choosing $2x-3=1$ immediately identifies the input needed to obtain the coefficient sum.

Enter $P(x)=-11(5x-8)^2+7(3x-4)^2+9(5x-8)(3x-4)$, then enter $P(2)$. Desmos shows the value of the original expression at the needed input.

Enter $18P(2)$ to verify the numerator of the rewritten expression when $2x-3=1$.

In the rewritten form, make $2x-3=1$. Then $(2x-3)^2=1$ also, so the expression becomes $\frac{a+b+c}{18}$. Solving $2x-3=1$ gives $x=2$.

At $x=2$, both $5x-8$ and $3x-4$ equal $2$. Therefore, $P(2)=-11(2)^2+7(2)^2+9(2)(2)=-44+28+36=20$.

Because $P(2)=\frac{a+b+c}{18}$, multiply by $18$: $a+b+c=18(20)=360$.