The problem asks us to multiply and simplify the expression $(4\sqrt{3} + \sqrt{2})(\sqrt{3} - 5\sqrt{2})$. We are to assume that all variables represent nonnegative numbers.

1. Problem Description

The problem asks us to multiply and simplify the expression $(4\sqrt{3} + \sqrt{2})(\sqrt{3} - 5\sqrt{2})$. We are to assume that all variables represent nonnegative numbers.

2. Solution Steps

We will use the distributive property (FOIL method) to expand the expression.

$(4\sqrt{3} + \sqrt{2})(\sqrt{3} - 5\sqrt{2}) = (4\sqrt{3})(\sqrt{3}) + (4\sqrt{3})(-5\sqrt{2}) + (\sqrt{2})(\sqrt{3}) + (\sqrt{2})(-5\sqrt{2})$

$= 4(\sqrt{3} \cdot \sqrt{3}) - 20\sqrt{3 \cdot 2} + \sqrt{2 \cdot 3} - 5(\sqrt{2} \cdot \sqrt{2})$

$= 4(3) - 20\sqrt{6} + \sqrt{6} - 5(2)$

$= 12 - 20\sqrt{6} + \sqrt{6} - 10$

Combine like terms:

$= 12 - 10 - 20\sqrt{6} + \sqrt{6}$

$= 2 - 19\sqrt{6}$

3. Final Answer

$2 - 19\sqrt{6}$