#### Answer

The solutions are $x=-1$ and $x=\pm\dfrac{1}{3}$

#### Work Step by Step

$x+1=9x^{3}+9x^{2}$
Take all terms to the right side of the equation:
$0=9x^{3}+9x^{2}-x-1$
Rearrange:
$9x^{3}+9x^{2}-x-1=0$
Group the first two terms together and the last two terms together:
$(9x^{3}+9x^{2})-(x+1)=0$
Take out common factor $9x^{2}$ from the first group:
$9x^{2}(x+1)-(x+1)=0$
Factor out $(x+1)$:
$(x+1)(9x^{2}-1)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x+1=0$
$x=-1$
$9x^{2}-1=0$
$9x^{2}=1$
$x^{2}=\dfrac{1}{9}$
$\sqrt{x^{2}}=\pm\sqrt{\dfrac{1}{9}}$
$x=\pm\dfrac{1}{3}$
The solutions are $x=-1$ and $x=\pm\dfrac{1}{3}$