maths olympiad type

let us assume thatn^{3}+an^{2}+bn+c is a sqaure for all n>0.

for sufficiently large `n` ,

\left(n^{\frac{3}{2}}+\frac{1}{2}an^{\frac{1}{2}}-1 \right)^{2}<(^{3}n+an^{2}+bn+c)<(n^{\frac{3}{2}}+\frac{1}{2}an^{\frac{1}{2}+1})^{2}

so if `n` is a lrge even perfect square, we have n^{3}+an^{2}+bn+c=\left(n^{\frac{3}{2}}+\frac{1}{2}an^{\frac{1}{2}} \right)^{2}

but RHS is not a perfect square for `n ` which is an even non square

edit : in that step with 2 simultaneous inequalities, in the extreme right , the term is \left(n^{3}+\frac{1}{2}an^{\frac{1}{2}}+1\right)^{2}