assume that all numbers represent real numbers

Question

(3√2-2√5)(4√2+√5)

Answer 1

We need to use the distributive property to expand out this product of two sums...in general

 

(A+B)(C+D)=AC+AD+BC+BD

 

The acronym FOIL meaning FIRST, OUTER, INNER, and LAST correspond to the four terms in the expanded product...

 

AB is the product of the first terms, AD is the product of the outer terms, BC is the product of the inner terms, and BD is the product of the last terms.

 

For your problem above, 

lets use x=\sqrt(2) and y=\sqrt(5),

 

(3x-2y)(4x+y) = 12x^2+3xy-8xy-2y^2 = 12x^2 - 5xy - 2y^2

 

x^2 = 2 and y^2=5, and xy=\sqrt(2)*\sqrt(5) = \sqrt(10)

 

so we have 12*2 - 5*(\sqrt(10)) - 2*5 = 14 - 5√10