error analysis - Has my textbook given the incorrect equation for calculating uncertainty in multiplication?

I am 99% sure that the calculation to obtain the uncertainty of two multiplied values given by my textbook and this university (http://web.uvic.ca/~jalexndr/192UncertRules.pdf) is incorrect.

They both say this:

$$(A \pm a)(B \pm b) = AB \pm (\varepsilon_A + \varepsilon_B)$$

Which should be equivalent to (correct me if I'm wrong):

$$(A \pm a)(B \pm b) = AB \pm (AB (\frac{a}{A} + \frac{b}{B})) = AB \pm (Ba + Ab))$$

Lets consider: $(4 \pm 1)(2 \pm 1)$ This should equal (8 \pm 6) right?

meaning the smallest value possible is 2 while the largest is 14. However we can see that the largest possible value is actually 15 (5 * 3) and the smallest is 3 (3 * 1). If this is plot on a 3D graph you will see there is no way to obtain lower or larger values than 3 and 15. Therefore there must be a problem with the formula. After some math I came up with the following equation which I believe is the correct equation:

$$(A \pm a)(B \pm b) = (AB + ab) ± (Ab + Ba))$$

As long as $A ≥ a$ and $B ≥ b$ and $a, b ≥ 0$.

Is the equation in the textbook (and given by this university) incorrect or have I just missed something?