You can multiply the brackets using the distributive law, in exactly the same way as in cases such as:
(x+3)(x-4) = x^2 -4x +3x-12(x+3)(x−4)=x2−4x+3x−12
Each term in the first bracket must be multiplied by each term is the second bracket.
(color(red)(5sqrt2) +color(blue)(3sqrt5))(2sqrt10-5)(5√2+3√5)(2√10−5)
=color(red)(5sqrt2)(2sqrt10-5) +color(blue)(3sqrt5)(2sqrt10-5)=5√2(2√10−5)+3√5(2√10−5)
=10sqrt20-25sqrt2+6sqrt50-15sqrt5=10√20−25√2+6√50−15√5
Now find factors for the roots, using squares where possible:
=10sqrt(color(magenta)(4)xx5)-25sqrt2+6sqrt((color(lime)25xx2)-15sqrt5=10√4×5−25√2+6√(25×2)−15√5
=10xxcolor(magenta)(2)sqrt5-25sqrt2+6 xxcolor(lime)5sqrt2 -15sqrt5=10×2√5−25√2+6×5√2−15√5
=20sqrt5 -15sqrt5+30sqrt2-25sqrt2=20√5−15√5+30√2−25√2
=5sqrt5+5sqrt2=5√5+5√2
=5(sqrt5 +sqrt2)=5(√5+√2)
