Let's first expand and simplify the given expression to make sure we're on the same page.

Given Expression:

$\left(5\sqrt{2} + \sqrt{3}\right)\left(2\sqrt{2} - 3\sqrt{3}\right)$

Expansion:

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

Simplification:

1. $(5\sqrt{2})(2\sqrt{2}) = 10 \times 2 = 20$
2. $(5\sqrt{2})(-3\sqrt{3}) = -15\sqrt{6}$
3. $(\sqrt{3})(2\sqrt{2}) = 2\sqrt{6}$
4. $(\sqrt{3})(-3\sqrt{3}) = -9$

Combine Like Terms:

$= 20 - 15\sqrt{6} + 2\sqrt{6} - 9$ $= (20 - 9) + (-15\sqrt{6} + 2\sqrt{6})$ $= 11 - 13\sqrt{6}$

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Now that we have the simplified result, here are 5 similar problems for you to solve:

1. $\left(3\sqrt{5} + 2\sqrt{7}\right)\left(4\sqrt{5} - \sqrt{7}\right)$
2. $\left(2\sqrt{3} - 5\sqrt{2}\right)\left(3\sqrt{3} + 4\sqrt{2}\right)$
3. $\left(\sqrt{6} + 2\sqrt{2}\right)\left(\sqrt{6} - 3\sqrt{2}\right)$
4. $\left(4\sqrt{2} + \sqrt{5}\right)\left(2\sqrt{2} - \sqrt{5}\right)$
5. $\left(7\sqrt{3} - \sqrt{2}\right)\left(3\sqrt{3} + 2\sqrt{2}\right)$

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Tip: When multiplying binomials with square roots, treat $\sqrt{a}$ like a variable during expansion, but remember to simplify square roots after multiplying (e.g., $\sqrt{2} \times \sqrt{2} = 2$).

Let me know if you need help solving any of these or want further explanations!