Expand the expression (sqrt(5) + sqrt(13))² using the formula (a+b)² = a² + 2ab + b².

Here, $a = \sqrt{5}$ and $b = \sqrt{13}$.

$$(\sqrt{5} + \sqrt{13})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{13}) + (\sqrt{13})^2$$

$$(\sqrt{5})^2 = 5$$ $$(\sqrt{13})^2 = 13$$ $$2(\sqrt{5})(\sqrt{13}) = 2\sqrt{5 \times 13} = 2\sqrt{65}$$

$$(\sqrt{5} + \sqrt{13})^2 = 5 + 2\sqrt{65} + 13$$ $$(\sqrt{5} + \sqrt{13})^2 = 18 + 2\sqrt{65}$$

Subtract 18 from both sides of the inequality:

$$2\sqrt{65} > 34 - 18$$ $$2\sqrt{65} > 16$$

$$\sqrt{65} > \frac{16}{2}$$ $$\sqrt{65} > 8$$

Since both sides are positive, the inequality direction remains the same.

$$(\sqrt{65})^2 > 8^2$$ $$65 > 64$$

Since $65 > 64$ is a true statement, the original inequality $(\sqrt{5} + \sqrt{13})^2 > 34$ is also true.

We have shown that $(\sqrt{5} + \sqrt{13})^2 > 34$.
$\text{Proven}$