The problem asks to multiply and simplify the expression $(\sqrt{5} + 8\sqrt{7x})(2\sqrt{5} + 6\sqrt{7x})$, assuming that the variable $x$ represents a non-negative number.

1. Problem Description

The problem asks to multiply and simplify the expression $(\sqrt{5} + 8\sqrt{7x})(2\sqrt{5} + 6\sqrt{7x})$, assuming that the variable $x$ represents a non-negative number.

2. Solution Steps

First, we use the distributive property (FOIL method) to expand the expression:

$(\sqrt{5} + 8\sqrt{7x})(2\sqrt{5} + 6\sqrt{7x}) = (\sqrt{5})(2\sqrt{5}) + (\sqrt{5})(6\sqrt{7x}) + (8\sqrt{7x})(2\sqrt{5}) + (8\sqrt{7x})(6\sqrt{7x})$

Next, we simplify each term:

$(\sqrt{5})(2\sqrt{5}) = 2(\sqrt{5})^2 = 2(5) = 10$

$(\sqrt{5})(6\sqrt{7x}) = 6\sqrt{5(7x)} = 6\sqrt{35x}$

$(8\sqrt{7x})(2\sqrt{5}) = 16\sqrt{7x(5)} = 16\sqrt{35x}$

$(8\sqrt{7x})(6\sqrt{7x}) = 48(\sqrt{7x})^2 = 48(7x) = 336x$

Now we combine all the terms:

$10 + 6\sqrt{35x} + 16\sqrt{35x} + 336x$

We can combine the terms with $\sqrt{35x}$:

$6\sqrt{35x} + 16\sqrt{35x} = 22\sqrt{35x}$

So the final expression is:

$10 + 22\sqrt{35x} + 336x$

3. Final Answer

$336x + 22\sqrt{35x} + 10$