Answer
$x=0.5$ or $x=-3$
Work Step by Step
We know that $a^b=a^c\longrightarrow b=c$ if $a\ne1,a\ne-1$ (which applies here).
Hence here $2^3=(2^2)^{x^2}2^{5x}\\2^3=2^{2x^2}2^{5x}\\2^3=2^{2x^2+5x}\\2x^2+5x=3\\2x^2+5x-3=(2x-1)(x+3)$
Thus $x=0.5$ or $x=-3$
