The given expression is:

$\sqrt{12} \left( 5\sqrt{3} + \sqrt{12} \right)$

Step-by-step Solution:

Step 1: Simplify $\sqrt{12}$

$\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}.$

Step 2: Substitute $\sqrt{12}$ in the expression

Replacing $\sqrt{12}$ with $2\sqrt{3}$, the expression becomes: $2\sqrt{3} \left( 5\sqrt{3} + 2\sqrt{3} \right).$

Step 3: Simplify the terms inside the parentheses

$5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}.$

So, the expression is now: $2\sqrt{3} \cdot 7\sqrt{3}.$

Step 4: Multiply the terms

$2\sqrt{3} \cdot 7\sqrt{3} = 2 \cdot 7 \cdot (\sqrt{3} \cdot \sqrt{3}).$

Since $\sqrt{3} \cdot \sqrt{3} = 3$, we have: $2 \cdot 7 \cdot 3 = 42.$

Final Answer:

$\boxed{42}$

Let me know if you would like further clarification or additional steps explained!

Questions for Practice:

1. How do you simplify radicals like $\sqrt{20}$?
2. What is the distributive property and how is it applied in this problem?
3. Can you expand $(a + b)(c + d)$ and simplify it?
4. What is the result of $(\sqrt{3})^2$?
5. How would the problem change if the coefficient of $\sqrt{3}$ were different?

Tip:

Always simplify radicals first to make your calculations easier!