How do we simplify this with multiplication?

Question

I have

$$(2a + 1)(2b + 1)$$

which apparently expands to

$$4ab + 2a + 2b + 1$$

and then can be written as

$$2(2ab + a + b) + 1 \,.$$

From where did we get the $4ab$ term?

Answer 1

For this, an application of the distributive law gives us that for any $a,b,c,d \in \mathbb{R}$ that $(a + b)(c + d) = (ac + db + bc + bd)$. In this case we would then have, $$(2a + 1)(2b + 1) = (2a)(2b) + (1)(2a) + (1)(2b) + (1)(1) = 4ab + 2a + 2b +1$$ as desired.