Answer
Radius of convergence is $\dfrac{9}{4}$
Work Step by Step
We need to apply the Ratio Test to the series.
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_n}|=\lim\limits_{n \to \infty} \dfrac{(2n+2)^2x}{(3n+2)^2}$
or, $=|x| \lim\limits_{n \to \infty}(\dfrac{2n+2}{3n+2})^2$
or, $=|x| \lim\limits_{n \to \infty}(\dfrac{2+2/n}{3+2/n})^2$
or, $=\dfrac{4}{9}|x|$
Now, $\dfrac{4}{9}|x|| \lt 1 \implies |x| \lt \dfrac{9}{4}$
Thus, the radius of convergence is $\dfrac{9}{4}$
