We need to simplify the expression:

$\left(\sqrt{10} + \sqrt{6}\right) \sqrt{8 - 2\sqrt{15}}$

Step 1: Simplify the expression under the square root

We first focus on simplifying $\sqrt{8 - 2\sqrt{15}}$. This expression is of the form $\sqrt{a - 2\sqrt{b}}$, which can sometimes be expressed as:

$\sqrt{a - 2\sqrt{b}} = \sqrt{x} - \sqrt{y}$

To apply this, we need $a = x + y$ and $b = xy$.

Finding $x$ and $y$:

• Here $a = 8$ and $b = 15$.
• We want to find $x$ and $y$ such that: $x + y = 8 \quad \text{and} \quad xy = 15$

These are the two equations we need to solve. Let’s solve them by factoring.

From the equation $x + y = 8$ and $xy = 15$, $x$ and $y$ must be the solutions to the quadratic equation:

$t^2 - (x + y)t + xy = 0$

Substituting $x + y = 8$ and $xy = 15$, we get:

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

Solving this quadratic equation using the quadratic formula:

$t = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(15)}}{2(1)}$

$t = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2} = \frac{8 \pm 2}{2}$

Thus, $t = 5$ and $t = 3$, so $x = 5$ and $y = 3$.

Step 2: Substitute and simplify

Now that we have $x = 5$ and $y = 3$, we can rewrite the square root as:

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

Step 3: Expand the original expression

We now substitute this into the original expression:

$\left(\sqrt{10} + \sqrt{6}\right)(\sqrt{5} - \sqrt{3})$

Now, expand using the distributive property:

$= \sqrt{10} \cdot \sqrt{5} - \sqrt{10} \cdot \sqrt{3} + \sqrt{6} \cdot \sqrt{5} - \sqrt{6} \cdot \sqrt{3}$

Simplifying each term:

$= \sqrt{50} - \sqrt{30} + \sqrt{30} - \sqrt{18}$

$= 5\sqrt{2} - \sqrt{18}$

Step 4: Simplify $\sqrt{18}$

We can simplify $\sqrt{18}$ as:

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Step 5: Final simplification

Substitute this back:

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

Thus, the simplified expression is:

$\boxed{2\sqrt{2}}$

Let me know if you have any questions or if you would like further details!

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Here are some related questions to practice:

1. Simplify $\sqrt{12 - 2\sqrt{35}}$.
2. Expand $(\sqrt{7} + \sqrt{5})(\sqrt{3} + \sqrt{2})$.
3. Solve $\sqrt{5 + 2\sqrt{6}}$ in its simplified form.
4. Simplify $\sqrt{6 + 2\sqrt{8}}$.
5. Expand $(\sqrt{8} - \sqrt{6})(\sqrt{10} + \sqrt{2})$.

Tip: When simplifying expressions involving square roots, look for patterns of the form $\sqrt{a \pm 2\sqrt{b}}$, as they often simplify nicely.