Answer
False
Work Step by Step
For the equation to have solution, the right side must be nonnegative, so $x\leq 0$.
$$\begin{aligned}
\sqrt{x^2+9 x+3}&=-x\\
x^2+9x+3&=x^2\\
9x+3&=0\\
9x&=-3\\
x&=-\frac{1}{3}.
\end{aligned}$$
As $x<0$, the solution fits, so we found a solution, therefore the given statement is false.
