5^(2x) + 5^(3x-1) = 150
$\displaystyle 5^{2z}= (5^x)^2$ and $\displaystyle 5^{3x-1}= \frac{(5^x)^3}{5}$
Your equation is $\displaystyle (5^x)^2+ \frac{(5^x)^3}{5}= 150$. If you let $\displaystyle y= 5^x$ you have $\displaystyle y^2+ y^3/5= 150$ or the cubic polynomial $\displaystyle y^3+ 5y^2= 750$. Any rational solution for y must be a factor of 750. I suggest you whack your friend about the ears!