2006 AMC 12B Problem 12

The vertex form is $y = a(x - p)^2 + p.$ At $x = 0,$ $y = ap^2 + p = -p,$ so $ap^2 = -2p$ and $a = -\dfrac{2}{p}.$ Expanding, $y = a x^2 - 2ap\, x + ap^2 + p,$ so $b = -2ap = -2\left(-\dfrac{2}{p}\right)p = 4.$ Thus, the correct answer is D.