Factoring when multiplying rational functions

I'm curious why multiplying the numerators of two rational functions before factoring them results in an incorrect solution.

Suppose we need to find the product of the following expression in lowest terms:

$$ \frac{5x-10}{x^2-36}\cdot\frac{4x+24}{x^2+5x-14} $$

Traditionally, you'd factor the numerators and denominators and then simplify (keeping track of any domain exclusions):

$$ \frac{5(x-2)}{(x+6)(x-6)}\cdot\frac{4(x+6)}{(x+7)(x-2)} $$

Answer:

$$ \frac{20}{(x-6)(x+7)} $$

$$ x\neq -6, 2 $$

It seems to me that it should also be possible to multiply the numerators together first, and then factor them. If we do this with the above example we'd get:

$(5x-10)(4x+24)$

$20x^2+120x-40x-240$

$20x^2+80x-240$

$x^2+4-12$

$(x+6)(x-2)$

Plugging this factored numerator into the expression results in:

$$ \frac{(x+6)(x-2)}{(x+6)(x-6)(x+7)(x-2)} $$

This ultimately results in an incorrect answer of:

$$ \frac{1}{(x-6)(x+7)} $$ $$ x \neq -6, 2 $$

You factored out a 20 when you multiplied out the numerators. $20x^{2}+80x-240 = 20(x+6)(x-2)$