Multiply by the conjugate.

For the numerator: $(2\sqrt{2} + 3\sqrt{3})(5\sqrt{2} + 2\sqrt{3})$

$$(2\sqrt{2})(5\sqrt{2}) + (2\sqrt{2})(2\sqrt{3}) + (3\sqrt{3})(5\sqrt{2}) + (3\sqrt{3})(2\sqrt{3})$$ $$10(\sqrt{2}\sqrt{2}) + 4(\sqrt{2}\sqrt{3}) + 15(\sqrt{3}\sqrt{2}) + 6(\sqrt{3}\sqrt{3})$$ $$10(2) + 4\sqrt{6} + 15\sqrt{6} + 6(3)$$ $$20 + 19\sqrt{6} + 18$$ $$38 + 19\sqrt{6}$$

For the denominator: $(5\sqrt{2} - 2\sqrt{3})(5\sqrt{2} + 2\sqrt{3})$
This is in the form $(a-b)(a+b) = a^2 - b^2$.

$$(5\sqrt{2})^2 - (2\sqrt{3})^2$$ $$(5^2 \times (\sqrt{2})^2) - (2^2 \times (\sqrt{3})^2)$$ $$(25 \times 2) - (4 \times 3)$$ $$50 - 12$$ $$38$$

$$\frac{38}{38} + \frac{19\sqrt{6}}{38}$$ $$1 + \frac{\sqrt{6}}{2}$$