How do you simplify 2 sqrt(3) *times* (5 sqrt(2) + 5 sqrt(5))?

$2 \sqrt{3} \left(5 \sqrt{2} + 5 \sqrt{5}\right) = 10 \sqrt{6} + 10 \sqrt{15}$
You cannot add unlike square roots, but you can multiply them. The only thing to do here is to distribute $2 \sqrt{3}$ to the two terms inside the parentheses. So $2 \sqrt{3} \cdot 5 \sqrt{2} = \left(2 \cdot 5\right) \left(\sqrt{3 \cdot 2}\right) = 10 \sqrt{6}$
Likewise, $2 \sqrt{3} \cdot 5 \sqrt{5} = \left(2 \cdot 5\right) \left(\sqrt{3 \cdot 5}\right) = 10 \sqrt{15}$
$2 \sqrt{3} \left(5 \sqrt{2} + 5 \sqrt{5}\right) = 10 \sqrt{6} + 10 \sqrt{15}$