5. The number of non-negative integer solutions $(x, y)$ for the equation $3 x^{2}+x y+y^{2}=3 x-2 y$ is ( )
(A) 0
(B) 1
(C) 2
(D) 3
5.C.

From $3 x^{2}+x y+y^{2}=3 x-2 y$, we can obtain the equation about $y$: $y^{2}+(2+x) y+3 x^{2}-3 x=0$.
. Then $\Delta=(2+x)^{2}-4\left(3 x^{2}-3 x\right)$
$$
\begin{array}{l}
=-11 x^{2}+16 x+4 \\
\leqslant \frac{16^{2}+4 \times 11 \times 4}{4 \times 11} \leqslant 10,
\end{array}
$$

Therefore, when $-11 x^{2}+16 x+4=0$ or 1 or 4 or 9, the original equation has integer roots.
Upon inspection, we find $x=0$ or 1.
Thus, $(x, y)=(0,0)$ or $(1,0)$ or $(0,-2)$ or $(1,-3)$, but $(0,-2)$ and $(1,-3)$ do not meet the requirements.