To expand the expression $(3\sqrt{7} + 4)^2$, use the formula for the square of a binomial $(a + b)^2 = a^2 + 2ab + b^2$.

Let $a = 3\sqrt{7}$ and $b = 4$.

1. $a^2 = (3\sqrt{7})^2 = 9 \times 7 = 63$

2. $2ab = 2 \times 3\sqrt{7} \times 4 = 24\sqrt{7}$

3. $b^2 = 4^2 = 16$

Putting it together:

$(3\sqrt{7} + 4)^2 = 63 + 24\sqrt{7} + 16$

So the expanded expression is:

$79 + 24\sqrt{7}$