To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.
(color(red)(3sqrt(6)) + color(red)(2sqrt(10)))(color(blue)(2sqrt(2)) + color(blue)(3sqrt(5))) becomes:
(color(red)(3sqrt(6)) xx color(blue)(2sqrt(2))) + (color(red)(3sqrt(6)) xx color(blue)(3sqrt(5))) + (color(red)(2sqrt(10)) xx color(blue)(2sqrt(2))) + (color(red)(2sqrt(10)) xx color(blue)(3sqrt(5)))
(6color(red)(sqrt(6))color(blue)(sqrt(2))) + (9color(red)(sqrt(6))color(blue)(sqrt(5))) + (4color(red)(sqrt(10))color(blue)(sqrt(2))) + (6color(red)(sqrt(10))color(blue)(sqrt(5)))
We can next use this rule for radicals to simplify the radical terms:
sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))
(6sqrt(color(red)(6) * color(blue)(2))) + (9sqrt(color(red)(6) * color(blue)(5))) + (4sqrt(color(red)(10) * color(blue)(2))) + (6sqrt(color(red)(10) * color(blue)(5)))
6sqrt(12) + 9sqrt(30) + 4sqrt(20) + 6sqrt(50)
We can now rewrite the terms in the radicals and use the above rule in reverse to further simplify the terms:
6sqrt(4 * 3) + 9sqrt(30) + 4sqrt(4 * 5) + 6sqrt(25 * 2)
(6sqrt(4)sqrt(3)) + 9sqrt(30) + (4sqrt(4)sqrt(5)) + (6sqrt(25)sqrt(2))
(6 * 2sqrt(3)) + 9sqrt(30) + (4 * 2sqrt(5)) + (6 * 5sqrt(2))
12sqrt(3) + 9sqrt(30) + 8sqrt(5) + 30sqrt(2)
