The given mathematical expression is:

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

We will expand it using the distributive property $(a+b)(c+d) = ac + ad + bc + bd$.

Expansion Steps:

1. Multiply the first terms: $5\sqrt{2x} \cdot (-4\sqrt{2x}) = -20x$

2. Multiply the outer terms: $5\sqrt{2x} \cdot \sqrt{5x} = 5\sqrt{10x^2} = 5x\sqrt{10}$

3. Multiply the inner terms: $\sqrt{5} \cdot (-4\sqrt{2x}) = -4\sqrt{10x}$

4. Multiply the last terms: $\sqrt{5} \cdot \sqrt{5x} = \sqrt{25x} = 5\sqrt{x}$

Combine all terms:

$-20x + 5x\sqrt{10} - 4\sqrt{10x} + 5\sqrt{x}$

This is the expanded and simplified form of the expression:

$-20x + 5x\sqrt{10} - 4\sqrt{10x} + 5\sqrt{x}$

Let me know if you'd like a step-by-step explanation of any part of this process or have additional questions.

Related Questions:

1. How can we factor similar expressions back to their original form?
2. What are some common properties of square roots in algebra?
3. How can we handle square roots involving fractional exponents?
4. Can we simplify expressions involving terms like $\sqrt{10x}$ further?
5. How does this relate to polynomial multiplication?

Tip:

When dealing with square roots, always check if the radicand (the term inside the root) can be simplified for easier computation.