Let the coefficient of x r in the expansion of x + 3 n − 1 + x + 3 n − 2 x + 2 + x + 3 n − 3 x + 2 2 + . ...…

Let,

$S={\left(x+3\right)}^{n-1}+{\left(x+3\right)}^{n-2}\left(x+2\right)+{\left(x+3\right)}^{n-3}{\left(x+2\right)}^2+.....+{\left(x+2\right)}^{n-1}$

Now, using the sum of G.P formula we get,

$\Rightarrow S={\left(x+3\right)}^{n-1}\left[\frac{1-{\left(\frac{x+2}{x+3}\right)}^n}{1-\frac{x+2}{x+3}}\right]$

$\Rightarrow S={\left(x+3\right)}^n-{\left(x+2\right)}^n$

Now, finding the sum of coefficients we get,

$\Rightarrow \sum {\alpha }_r=\sum {\left(x+3\right)}^n-\sum {\left(x+2\right)}^n$

$\Rightarrow 4^n-3^n={\beta }^n-{\gamma }^n$ $\left\{\text{by taking }x=1\right\}$

So, on comparing both side we get, $\beta =4, \gamma =3$

Hence, ${\beta }^2+{\gamma }^2=16+9=25$