To simplify the expression:

$\frac{2}{7 + 5\sqrt{2}},$

we need to rationalize the denominator. To do this, we multiply both the numerator and denominator by the conjugate of the denominator, which is $7 - 5\sqrt{2}$:

$\frac{2}{7 + 5\sqrt{2}} \times \frac{7 - 5\sqrt{2}}{7 - 5\sqrt{2}} = \frac{2(7 - 5\sqrt{2})}{(7 + 5\sqrt{2})(7 - 5\sqrt{2})}.$

Now simplify the denominator using the difference of squares:

$(7 + 5\sqrt{2})(7 - 5\sqrt{2}) = 7^2 - (5\sqrt{2})^2 = 49 - 50 = -1.$

So the expression becomes:

$\frac{2(7 - 5\sqrt{2})}{-1} = -2(7 - 5\sqrt{2}) = -14 + 10\sqrt{2}.$

Thus, the simplified expression is:

$-14 + 10\sqrt{2}.$

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do you rationalize the denominator in complex expressions?
2. Why is it useful to use the conjugate when rationalizing?
3. What are the properties of square roots in rationalizing fractions?
4. How can the difference of squares formula simplify rationalization?
5. Can you simplify $\frac{3}{4 + \sqrt{3}}$ using a similar process?

Tip: Always use the conjugate when rationalizing expressions with irrational numbers in the denominator.