If we complete the square for $f(X)$ above (where $f \in \mathbb{Z}[X]$ is the field of polynomials with integer coefficients), we obtain: $a\left(X^2 +\frac{b}{a}X\right) + c \Rightarrow a\left(X+\frac{b}{2a}\right)^2 + \left(c-\frac{b^2}{4a}\right)$ (i). Now, if $f(n)$ is a perfect square for all $n \in \mathbb{N}$, then we require: $a = N^2$ (ii); $2a | b \Rightarrow b = k \cdot 2a = 2k \cdot N^2$ (iii); $c - \frac{b^2}{4a} = 0 \Rightarrow c = \frac{b^2}{4a} = \frac{4k^2N^4}{4N^2} = k^2N^2$ (iv). for some $N, k \in \mathbb{Z}$. Substitution of (ii) through (iv) into $f$ yields: $f(X)= aX^2+bX+c = N^2X^2 + 2kN^2X + k^2N^2 = N^2(X^2 + 2kX + k^2) = N^2(X+k)^2 =g^{2}(X)$ for polynomial $g(X) \in \mathbb{Z}[X]$. QED