That helped tremendously. I found a to equal 0.25Easier to work with it in factorised form.

Because it is a fourth degree polynomial with factors of (x-2) raised to an even power and (x+4) raised to an even power then

\(\displaystyle f(x) = a(x-2)^2(x+4)^2\) to give a fourth degree polynomial (when expanded out).

Your job now is to find a such that the graph goes through (0, 16).