so we have

(sqrt(3) - sqrt(-4))(sqrt(6) - sqrt(-8))

ok, we will expand these brackets just as we would normally, that is, we take the first term in the first set of brackets and multiply all the terms in the second set of brackets, then we take the second term in the first set of brackets and multiply everything in the second pair of brackets. but first, let's "simplify" the square roots of the negative numbers.

we know sqrt(-4) = sqrt(4)*sqrt(-1), a law of surds says we can split up square roots like that, and a law of exponents as well, but sqrt(-1) = i

so sqrt(-4) = sqrt(4)*sqrt(-1) = 2i

similarly, sqrt(-8) = sqrt(8) i, now back to the problem

(sqrt(3) - sqrt(-4))(sqrt(6) - sqrt(-8))

= (sqrt(3) - 2i)(sqrt(6) - sqrt(8)i)

= sqrt(3)*sqrt(6) - sqrt(3)*sqrt(8)*i - 2i*sqrt(6) + 2*sqrt(8)*i^2 ....but i^2 = -1

so that is:

sqrt(3)*sqrt(6) - sqrt(3)*sqrt(8)*i - 2i*sqrt(6) - 2*sqrt(8) .......now to simplify

= sqrt(18) - 2sqrt(8) - sqrt(24)i - 2sqrt(6)i

= sqrt(2)*sqrt(9) - 2*sqrt(2)*sqrt(4) - sqrt(4)*sqrt(6)i - 2sqrt(6)i

= 3sqrt(2) - 4sqrt(2) - 2sqrt(6)i - 2sqrt(6)i

= -sqrt(2) - 4sqrt(6)i

now these steps can get a little confusing, you have to keep track of when i'm splitting square roots and when i'm recombining. if you miss a step or don't understand something, just tell me