$x - 3\sqrt {x + 1} + 3 > 0\Leftrightarrow x=3\sqrt {x + 1} + 3 > 0$
$\Leftrightarrow
x+3>3\sqrt{x+1}\Leftrightarrow\begin{cases} x\geq -1\\ (x+3)^2>9(x+1)\end{cases}$
$\Leftrightarrow
\begin{cases} x\geq -1\\ x^2-3x>0\end{cases}\Leftrightarrow\left[\begin{array}{I}
–1\leq x<0\\ x>3\end{array}\right.$