Determine constant c so that g(x,y) is continuous at every point $$g(x,y)=\begin{cases} \frac{x^3+xy^2+2x^2+2y^2}{x^2+y^2} & \text{if} & (x,y) \neq (0,0) \\
c & \text{if} & (x,y) = (0,0) \end{cases}$$
Should I set the first function equal to c and then solve using polar coordinates?

Hint. By using polar coordinates one gets, for $(x,y)\ne (0,0)$,
$$
g(x,y)=\frac{x^3+xy^2+2x^2+2y^2}{x^2+y^2}=\frac{r^3\cos^3 \theta+r^3\cos\theta \sin^2\theta +2r^2}{r^2},\quad r\ne0,
$$ that is
$$
g(x,y)=r\cos^3 \theta+r\cos\theta \sin^2\theta +2,\quad r\ne0,
$$ then this tends to $2$ as $r \to 0^+$.
Can you take it from here?
