Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x)*(b + g(x))$?
When f(x) multiplies g(x) there will be a term 2x8. No matter what size the constant b is, it won't affect the x8 term. Hence the degree of the resulting polynomial will be 8.