Troubles with solving $\sqrt{2x+3}-\sqrt{x-10}=4$

Note that when you square something like $a\sqrt{b}$ you get $a^2b$.

Thus, you should get:

$\begin{align} x^2-6x+9 &= 64(x-10)\\ x^2-6x+9 &= 64x-640\\ x^2-70x+649 &= 0\\ (x-11)(x-59) &= 0\\ \therefore \boxed{x=11,59}. \end{align}$