One Factor Is Enough

The rule says OR, and the or matters. In (x - 4)(x + 6) = 0, take x = 4: the first factor is 0, the second is 10, and 0 * 10 = 0. The second factor did not need to vanish; one zero kills the whole product.

So each case stands alone. Case one pretends the first factor is the zero and solves it; case two pretends it is the second. The answers 4 and -6 each make the ORIGINAL equation true separately, and a quick substitution shows it.

Do not hunt for an x that zeroes both factors at once; usually none exists, and none is needed. The solution set collects every x that works by ANY case: that is what or means.