(2sqrt5-4sqrt6)(3sqrt3+8sqrt2)
Expand using the FOIL method.
(2sqrt5-4sqrt6)(3sqrt3+8sqrt2)=
color(blue)((2sqrt5*3sqrt3))+color(teal)((2sqrt5*8sqrt2))+color(red)((-4sqrt6*3sqrt3))+color(green)((-4sqrt6*8sqrt2)
Multiply integers and square roots.
color(blue)((2*3*sqrt5sqrt3))+color(teal)((2*8*sqrt5sqrt2))+color(red)((-4*3*sqrt6sqrt3))+color(green)((-4*8*sqrt6sqrt2))
Simplify.
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-12sqrt18))+color(green)((-32sqrt12))
The first two sets of parentheses are cannot be simplified further. The second two sets of parentheses need further simplification.
Prime factorize color(red)((-12sqrt18)).
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-12sqrt(2*3*3))+color(green)((-32sqrt12)
Simplify.
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-12*3sqrt2))+color(green)((-32sqrt12))
Simplify.
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-36sqrt2))+color(green)((-32sqrt12))
Prime factorize color(green)((-32sqrt12)).
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-36sqrt2))+color(green)((-32sqrt(2*2*3)))
Simplify.
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-36sqrt2))+color(green)((-32*2sqrt3))
Simplify.
color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-36sqrt2))+color(green)((-64sqrt3))
Simplify.
6sqrt15+16sqrt10-36sqrt2-64sqrt3
