<s>[INST] Show that $x^4+x^3+x^2+x+1$ and $x^4+2x^3+2x^2+2x+5$ cannot be a square when $x\neq3$ and $x\neq2$ respectively. As the title says, I need help showing that $x^4+x^3+x^2+x+1$ and $x^4+2x^3+2x^2+2x+5$  cannot be a square when $x\neq3$  and $x\neq2$  respectively, where $x$ is a natural number.
In order to do this, I have to use the fact that if $x$ is any natural number >1, any natural number $n\geq x$ can be represented uniquely in the form
$n=c_0+c_1x+c_2x^2+...+c_mx^m$,
where $0\leq c_i \leq x-1$ for $i=0,1,2,...,m-1$  and
$0< c_m \leq x-1$.
 [/INST] Hint: IF either of these numbers were squares, then you could express them as $N^2$, where $N$ is some number. By what you know, you can write $N=c_0+c_1x+c_2x^2+\ldots+c_mx^m$ uniquely. Notice that $N$ uses at most three powers of $x$: $N=c_0+c_1x+c_2x^2$, since when you square it you need an $x^4$ to appear as the highest power. Now square $N$ out and show you can't have nonnegative $c_i$ which give you the coefficients of your numbers. For example, for $x^4+x^3+x^2+x+1$, you immediately get $c^2_0\equiv 1\pmod{x}$. 
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