In the \(xy\)-plane, if the point \((a,b)\) lies on the parabola with equation \(y=x^2+4\), which of the following must be equal to \(a^4\)?

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Since \((a,b)\) lies on the parabola,

$$ y=x^2+4 $$ $$ b=a^2+4 $$

We want to manipulate the equation so that we obtain \(a^4\) in terms of other terms and constants.

$$ b-4=a^2 $$ $$ (b-4)^2=a^4 $$ $$ \boxed{b^2-8b+16}=a^4 $$