Answer
$x=\displaystyle \frac{2}{5}$
Work Step by Step
We solve by factoring with grouping:
$5x^{3}+45x=2x^{2}+18$
$5x^{3}+45x-2x^{2}-18=0$
$5x^{3}-2x^{2}+45x-18=0$
$x^{2}(5x-2)+9(5x-2)=0$
$(5x-2)(x^{2}+9)=0$
$5x-2=0$ or $x^{2}+9=0$
$5x=2$ or $x^{2}=-9$
$x=\displaystyle \frac{2}{5}$ or $x=\pm \sqrt{-9}=$ not real
$x=\displaystyle \frac{2}{5}$