Let's simplify each term individually and then combine them:

Step 1: Simplify $3\sqrt{125}$

1. Break down $\sqrt{125}$: $\sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5}$
2. Substitute back: $3\sqrt{125} = 3 \cdot 5\sqrt{5} = 15\sqrt{5}$

Step 2: Simplify $4\sqrt{12}$

1. Break down $\sqrt{12}$: $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$
2. Substitute back: $4\sqrt{12} = 4 \cdot 2\sqrt{3} = 8\sqrt{3}$

Step 3: Simplify $3\sqrt{48}$

1. Break down $\sqrt{48}$: $\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$
2. Substitute back: $3\sqrt{48} = 3 \cdot 4\sqrt{3} = 12\sqrt{3}$

Step 4: Combine all terms

Now we have: $15\sqrt{5} - 8\sqrt{3} + 12\sqrt{3}$ Combine the terms with $\sqrt{3}$: $= 15\sqrt{5} + (12 - 8)\sqrt{3} = 15\sqrt{5} + 4\sqrt{3}$

Final Answer

$15\sqrt{5} + 4\sqrt{3}$

Would you like more details on any step or have any questions?

Here are 5 related questions to expand on this:

1. How do we simplify square roots of other non-perfect square numbers?
2. Can we add square root terms if they have different numbers under the square root?
3. What is the difference between simplifying square roots and rationalizing them?
4. How can this process be applied to roots of higher powers, like cube roots?
5. Why is it beneficial to break down the square root terms before combining?

Tip: Always look for perfect square factors to simplify square roots easily!