We are tasked with simplifying the expression:

$\left( 4\sqrt{3} \right)^2 - \left( 3\sqrt{2} + 5\sqrt{6} \right)^2$

Step 1: Simplify the square terms

First, square each term individually:

1. $\left( 4\sqrt{3} \right)^2 = 4^2 \times \left( \sqrt{3} \right)^2 = 16 \times 3 = 48$

2. To simplify $\left( 3\sqrt{2} + 5\sqrt{6} \right)^2$, we expand it using the binomial formula: $\left( 3\sqrt{2} + 5\sqrt{6} \right)^2 = \left( 3\sqrt{2} \right)^2 + 2 \times 3\sqrt{2} \times 5\sqrt{6} + \left( 5\sqrt{6} \right)^2$ Let's break this down:

   • $\left( 3\sqrt{2} \right)^2 = 9 \times 2 = 18$
   • $\left( 5\sqrt{6} \right)^2 = 25 \times 6 = 150$
   • The middle term: $2 \times 3\sqrt{2} \times 5\sqrt{6} = 2 \times 15 \times \sqrt{12} = 30 \times 2\sqrt{3} = 60\sqrt{3}$

Thus, the expansion of $\left( 3\sqrt{2} + 5\sqrt{6} \right)^2$ is: $18 + 150 + 60\sqrt{3} = 168 + 60\sqrt{3}$

Step 2: Combine everything

Now, we subtract the two expressions: $\left( 4\sqrt{3} \right)^2 - \left( 3\sqrt{2} + 5\sqrt{6} \right)^2 = 48 - \left( 168 + 60\sqrt{3} \right)$ Distribute the negative sign: $48 - 168 - 60\sqrt{3} = -120 - 60\sqrt{3}$

Thus, the simplified expression is: $\boxed{-120 - 60\sqrt{3}}$

Would you like any further clarifications or details?

Related questions:

1. How do you expand binomials using the distributive property?
2. What is the general method for squaring binomials involving radicals?
3. How can you simplify expressions with square roots?
4. What are some common mistakes when working with radical expressions?
5. How would you solve similar problems with cube roots?

Tip: Always double-check the middle term when expanding binomials with radicals, as it can easily be overlooked.