• Realizing the function we are finding has to be quadratic.
• Expressing the function as $ax^2+bx+c$ and finding $a,b,c$

$(2x^2-2x+2)^2 = 4x^4-8x^3+12x^3-8x+4$

Now, $2x^2-2x+2$ is always integer for integer $x$. So, $f(x)$ is always a perfect square.

Squaring a function with degree $n$ gives a function with degree $2n$ and the given is a quartic function. So, searching for a function with degree $2$ as $ax^2+bx+c$ is the best option.

$(ax^2+bx+c)^2 = a^x+2abx^3+(2ac+b^2)x^2+2bcx+c^2$. 

Comparing the coefficients, we get $a=2,c=2$ and $b=-2$