The same way you would multiply non-radicals. Break it down into simpler steps.
The front numbers multiply, then the outer, then the inner, then the last numbers. You might have learned this as "FOIL".
= (4sqrt5 * 2sqrt5) + (4sqrt5 * -4sqrt2) + (3sqrt5 * 2sqrt5) + (3sqrt5 * -4sqrt2)=(4√5⋅2√5)+(4√5⋅−4√2)+(3√5⋅2√5)+(3√5⋅−4√2)
= (8sqrt5*sqrt5) + (-16sqrt5 * sqrt2) + (6sqrt5 * sqrt5) + (-12sqrt5 * sqrt2)=(8√5⋅√5)+(−16√5⋅√2)+(6√5⋅√5)+(−12√5⋅√2)
= 8sqrt5*sqrt5 - 16sqrt5 * sqrt2 + 6sqrt5 * sqrt5 - 12sqrt5 * sqrt2=8√5⋅√5−16√5⋅√2+6√5⋅√5−12√5⋅√2
= 40 - 16sqrt10 + 30 - 12sqrt10=40−16√10+30−12√10
= color(blue)(70 - 28sqrt10)=70−28√10
Alternatively you could write this as:
= 7(sqrt10sqrt10 - 4sqrt10)=7(√10√10−4√10)
= 7(2sqrt5color(green)(sqrt5) - 4sqrt2color(green)(sqrt5))=7(2√5√5−4√2√5)
= color(blue)(7sqrt5(2sqrt5 - 4sqrt2))=7√5(2√5−4√2)
