Find all positive integer roots of the equation $\displaystyle $1 + n + n^2 + n^3 = A^2$$

My solution attempt:

Following Wolframalpha factoring, it can be rewritten as $\displaystyle (n+\frac{1}{3})^3+\frac{2}{3}(n+\frac{1}{3})+\frac {20}{27}=A^2$$, then

$\displaystyle $\displaystyle{(3n+1)^3+6(3n+1)+20=27A^2$$ and putting $\displaystyle $3n+1=k \Rightarrow n=\frac{k-1}{3}}$$ and for the

first positive integer root, $\displaystyle $k_1=4 \Rightarrow n_1=1 \Rightarrow A_1=2$$

second positive integer root, $\displaystyle $k_2=22 \Rightarrow n_2=7 \Rightarrow A_2=20$$

How can I prove that these $\displaystyle $[(n, A)=(1, 2),(7, 20)]$$ are the only positive integer solutions? Thanks for any suggestions. Regards.