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Let's solve the given problem step by step using the Distributive Property.

We have:

\[
(3x^2 + 7x + 6)(6x^2 - 5x + 10)
\]

### Step 1: Apply the Distributive Property
We will multiply each term in the first polynomial by every term in the second polynomial.

#### 1. Multiply \(3x^2\) by each term in the second polynomial:
\[
3x^2 \cdot 6x^2 = 18x^4
\]
\[
3x^2 \cdot (-5x) = -15x^3
\]
\[
3x^2 \cdot 10 = 30x^2
\]

#### 2. Multiply \(7x\) by each term in the second polynomial:
\[
7x \cdot 6x^2 = 42x^3
\]
\[
7x \cdot (-5x) = -35x^2
\]
\[
7x \cdot 10 = 70x
\]

#### 3. Multiply \(6\) by each term in the second polynomial:
\[
6 \cdot 6x^2 = 36x^2
\]
\[
6 \cdot (-5x) = -30x
\]
\[
6 \cdot 10 = 60
\]

### Step 2: Combine like terms
Now, let's combine the terms that have the same powers of \(x\).

\[
18x^4
\]
\[
(-15x^3 + 42x^3) = 27x^3
\]
\[
(30x^2 - 35x^2 + 36x^2) = 31x^2
\]
\[
(70x - 30x) = 40x
\]
\[
60
\]

### Final Answer:
\[
18x^4 + 27x^3 + 31x^2 + 40x + 60
\]

Would you like further clarification on any part of this process or any details on the properties used?

Here are five related questions to explore:
1. How does the distributive property work with higher-degree polynomials?
2. What is the difference between using the distributive property and FOIL in polynomial multiplication?
3. How do you simplify polynomials after multiplication?
4. Can the product of two polynomials be factored again?
5. How does polynomial multiplication relate to solving systems of equations?

**Tip**: Always double-check that you combine like terms correctly when performing polynomial multiplication.

Multiply the polynomials using the Distributive Property: (3x^2 + 7x + 6)(6x^2 - 5x + 10).

Learn how to multiply polynomials using the Distributive Property with a step-by-step solution. This example involves multiplying (3x^2 + 7x + 6) and (6x^2 - 5x + 10), covering key algebraic concepts like polynomial multiplication and combining like terms. Suitable for Grades 9-12.

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