jmc

First, we move all terms to one side to get $5x^2-3x-5=0.$ Seeing that factoring will not work, we apply the Quadratic Formula: $$\begin{array}{rl}x& =\frac{-(-3)\pm \sqrt{(-3{)}^2-4(5)(-5)}}{2(5)}\\ & =\frac{3\pm \sqrt{9+100}}{10}=\frac{3\pm \sqrt{109}}{10}.\end{array}$$Now we see that $10x=3\pm \sqrt{109}$, so $(10x-3{)}^2=\boxed{109}.$

Alternatively, from the equation $5x^2-3x-5=0$, $5x^2-3x=5$. Then $(10x-3{)}^2=100x^2-60x+9=20(5x^2-3x)+9=20\cdot 5+9=\boxed{109}$.