#### Answer

$x=-\displaystyle \frac{5}{3}$
$x=-\displaystyle \frac{4}{3}$

#### Work Step by Step

$(3x+5)^{4}-(3x+5)^{3} =0$
We can substitute $y$ for the repeating term:
$y=3x+5$
The new equation is thus:
$y^{4}-y^{3} =0$
We factor and solve:
$y(y^{3}-1)=0$
$y(y-1)(y^{2}+y+1)=0$
$y=0$ or $y-1=0$ or $y^{2}+y+1=0$
$y=0$ or $y=1$
(The last equation $y^{2}+y+1=0$ has no real solutions because the discriminant is negative.)
We need to solve for $x$ given the solutions for $y$:
$y=3x+5$
$x=\frac{y-5}{3}$
If $y=0$, then $x=\frac{0-5}{3}=\frac{-5}{3}$
If $y=1$, then $x=\frac{1-5}{3}=\frac{-4}{3}$
So the solutions are: $x=-\displaystyle \frac{5}{3}$ and $x=-\displaystyle \frac{4}{3}$