To evaluate the given expression:

$\sqrt{5 + 2\sqrt{6}} + \sqrt{8 - 2\sqrt{15}}$

Step 1: Simplify each term separately.

First term: $\sqrt{5 + 2\sqrt{6}}$

Assume that $\sqrt{5 + 2\sqrt{6}} = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are numbers to be determined.

Squaring both sides:

$5 + 2\sqrt{6} = (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$

This gives two equations:

1. $a + b = 5$
2. $2\sqrt{ab} = 2\sqrt{6}$, which simplifies to $ab = 6$

From $a + b = 5$ and $ab = 6$, these are the conditions for factoring a quadratic equation. The roots of the equation $x^2 - (a+b)x + ab = 0$ are:

$x^2 - 5x + 6 = 0$

Factoring:

$(x - 3)(x - 2) = 0$

Thus, $a = 3$ and $b = 2$.

Therefore:

$\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}$

Second term: $\sqrt{8 - 2\sqrt{15}}$

Similarly, assume that $\sqrt{8 - 2\sqrt{15}} = \sqrt{c} - \sqrt{d}$, where $c$ and $d$ are numbers to be determined.

Squaring both sides:

$8 - 2\sqrt{15} = (\sqrt{c} - \sqrt{d})^2 = c + d - 2\sqrt{cd}$

This gives two equations:

1. $c + d = 8$
2. $-2\sqrt{cd} = -2\sqrt{15}$, which simplifies to $cd = 15$

From $c + d = 8$ and $cd = 15$, these are the conditions for factoring a quadratic equation. The roots of the equation $x^2 - (c+d)x + cd = 0$ are:

$x^2 - 8x + 15 = 0$

Factoring:

$(x - 5)(x - 3) = 0$

Thus, $c = 5$ and $d = 3$.

Therefore:

$\sqrt{8 - 2\sqrt{15}} = \sqrt{5} - \sqrt{3}$

Step 2: Add the simplified terms together

Now, we add the two simplified terms:

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

Simplifying:

$\sqrt{2} + \sqrt{5}$

Thus, the value of the given expression is:

$\boxed{\sqrt{2} + \sqrt{5}}$

Let me know if you need further clarifications or have any questions.

Here are 5 related questions:

1. How do you solve similar problems involving nested square roots?
2. What is the general method for simplifying square root expressions?
3. Can you apply the same technique for cubed roots?
4. How does factoring quadratic equations help in simplifying square roots?
5. What is the geometric interpretation of nested radicals?

Tip: When simplifying square roots involving sums or differences of radicals, assuming a form like $\sqrt{a} \pm \sqrt{b}$ and squaring both sides can be an efficient strategy.