jmc

We don't know $g(x),$ so we don't have an expression we can simply stick $-5$ in to get an answer. We do, however, know that $g(f(x))=2x^2+5x-3.$ So, if we can figure out what to put into $f(x)$ such that $-5$ is output, we can use our expression for $g(f(x))$ to find $g(-5).$ If $f(x)=-5,$ we have $3x-8=-5,$ so $x=1.$ Therefore, letting $x=1$ in $g(f(x))=2x^2+5x-3$ gives $$g(-5)=g(f(1))=2\cdot 1^2+5\cdot 1-3=\boxed{4}.$$