How do we minimize $\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right)$? Find the minimum value of the following function, where a and b are real
numbers.
\begin{align} f(x)&=\left(x+a+b\right)\left(x+a-b\right)\left(x-a+b\right)\left(x-a-b\right) \end{align}
Note: The solution should not contain any calculus methods.
So I conjugated the parenthesizes two by two, and I got:
\begin{align} f(x)&=\left(x^2-(a+b)^2\right)\left(x^2-(a-b)^2\right) \end{align}
Can somebody help me to take this problem further?

Hint:
$$(y-2c)(y-2d)=(y-(c+d))^2+4cd-(c+d)^2\ge?$$
