Let the coefficient of x ^ r in the expansion of (x + 3) ^ n - 1 +...

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,

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

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

Now, finding the sum of coefficients we get,

$⇒ ∑ {\alpha}_{r} = ∑ {\left(x + 3\right)}^{n} - ∑ {\left(x + 2\right)}^{n}$

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

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

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