<s>[INST] Given that $n^4-4n^3+14n^2-20n+10$ is a perfect square, find all integers n that satisfy the condition So, I tried solving that by 
$$n^4-4n^3+14n^2-20n+10=x^2\\10=x^2-a^2, a^2=n^4-4n^3+14n^2-20n+10\\10=(x+a)(x-a)$$ but I couldn't find any integers when I solved it
 [/INST] It always helps to form squares from the biggest power and is a good strategy:
$$n^4-4n^3+14n^2-20n+10=n^2(n^2-4n+4)+10n^2-20n+10=\\
n^2(n-2)^2+10(n-1)^2=(n(n-2))^2+10(n-1)^2=((n-1)^2-1)^2+10(n-1)^2=\\
(n-1)^4-2(n-1)^2+1+10(n-1)^2=(n-1)^4+8(n-1)^2+1=\\
((n-1)^2+4)^2-15=x^2$$
I think this quite large hint will make it a bit easier to solve it.
Just as a note: it just happens we can make a nice square, that is not always the case!
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