To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.
(color(red)(sqrt(3)) - color(red)(sqrt(2)))(color(blue)(sqrt(15)) + color(blue)(sqrt(12))) becomes:
(color(red)(sqrt(3)) xx color(blue)(sqrt(15))) + (color(red)(sqrt(3)) xx color(blue)(sqrt(12))) - (color(red)(sqrt(2)) xx color(blue)(sqrt(15))) - (color(red)(sqrt(2)) xx color(blue)(sqrt(12)))
Next, use this rule of radicals to multiply the four sets of radicals:
sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))
sqrt(3 xx 15) + sqrt(3 xx 12) - sqrt(2 xx 15) - sqrt(2 xx 12)
sqrt(45) + sqrt(36) - sqrt(30) - sqrt(24)
sqrt(45) + 6 - sqrt(30) - sqrt(24)
Now, we can rewrite the expression to simplify as:
sqrt(9 xx 5) + 6 - sqrt(30) - sqrt(4 xx 6)
(sqrt(9) xx sqrt(5)) + 6 - sqrt(30) - (sqrt(4) xx sqrt(6))
3sqrt(5) + 6 - sqrt(30) - 2sqrt(6)
