jmc

We proceed as follows: $$\begin{array}{rl}(x-2)(2x+5)& =8x-6\\ 2x^2+x-10& =8x-6\\ 2x^2-7x-4& =0\\ (x-4)(2x+1)& =0.\end{array}$$This gives us $x=4$ or $x=-\frac{1}{2}.$ However, since the latter would make $x-2$ negative, we see that $x=\boxed{4}.$