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.

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