Let's simplify the expression $(4 \sqrt{7} + 3)(6 \sqrt{7} - 1)$ by applying the distributive property (also known as the FOIL method for binomials).

Step-by-step solution:

$(4 \sqrt{7} + 3)(6 \sqrt{7} - 1)$

We will distribute each term from the first binomial to each term in the second binomial:

1. Multiply $4 \sqrt{7}$ by $6 \sqrt{7}$: $4 \sqrt{7} \times 6 \sqrt{7} = 24 \times 7 = 168$

2. Multiply $4 \sqrt{7}$ by $-1$: $4 \sqrt{7} \times -1 = -4 \sqrt{7}$

3. Multiply $3$ by $6 \sqrt{7}$: $3 \times 6 \sqrt{7} = 18 \sqrt{7}$

4. Multiply $3$ by $-1$: $3 \times -1 = -3$

Now, add all these results together: $168 - 4 \sqrt{7} + 18 \sqrt{7} - 3$

Combine like terms:

$168 - 3 = 165$ $-4 \sqrt{7} + 18 \sqrt{7} = 14 \sqrt{7}$

Thus, the simplified expression is:

$165 + 14 \sqrt{7}$

Final Answer:

$165 + 14 \sqrt{7}$

Let me know if you'd like any further details or have additional questions! Here are some related questions for further learning:

1. How would you expand and simplify $(x + 2)(x - 3)$?
2. What happens when you apply the distributive property to $(a + b)(c + d)$?
3. How do you simplify expressions involving square roots, like $\sqrt{7} + \sqrt{3}$?
4. What is the significance of the FOIL method in binomial multiplication?
5. How can the distributive property be used in solving algebraic equations?

Tip: Always combine like terms after distributing to ensure the expression is fully simplified.