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(5)) - color(red)(sqrt(6)))(color(blue)(sqrt(5)) + color(blue)(sqrt(2))) becomes:
(color(red)(sqrt(5)) xx color(blue)(sqrt(5))) + (color(red)(sqrt(5)) xx color(blue)(sqrt(2))) - (color(red)(sqrt(6)) xx color(blue)(sqrt(5))) - (color(red)(sqrt(6)) xx color(blue)(sqrt(2)))
We can now use this rule for radicals to multiply the radicals:
sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))
sqrt(color(red)(5) xx color(blue)(5)) + sqrt(color(red)(5) xx color(blue)(2)) - sqrt(color(red)(6) xx color(blue)(5)) - sqrt(color(red)(6) xx color(blue)(2))
sqrt(25) + sqrt(10) - sqrt(30) - sqrt(12)
5 + sqrt(10) - sqrt(30) - sqrt(12)
We can then use this rule for radicals to simplify the term on the right:
sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))
5 + sqrt(10) - sqrt(30) - sqrt(12)
5 + sqrt(10) - sqrt(30) - sqrt(color(red)(4) * color(blue)(3))
5 + sqrt(10) - sqrt(30) - (sqrt(color(red)(4)) * sqrt(color(blue)(3)))
5 + sqrt(10) - sqrt(30) - 2sqrt(3)
