https://file.aiquickdraw.com/mathbot/file/1726680181165b3kenrm2.jpg

Let's solve the given expression step by step using distribution.

The given expression is:
\[
(4p + 2 + q)(p^2 - 3q + 6)
\]

We will use the distributive property, multiplying each term from the first binomial by each term from the second binomial.

1. **Multiply \(4p\) by each term in \((p^2 - 3q + 6)\):**

\[
4p \cdot p^2 = 4p^3
\]
\[
4p \cdot (-3q) = -12pq
\]
\[
4p \cdot 6 = 24p
\]

2. **Multiply \(2\) by each term in \((p^2 - 3q + 6)\):**

\[
2 \cdot p^2 = 2p^2
\]
\[
2 \cdot (-3q) = -6q
\]
\[
2 \cdot 6 = 12
\]

3. **Multiply \(q\) by each term in \((p^2 - 3q + 6)\):**

\[
q \cdot p^2 = p^2q
\]
\[
q \cdot (-3q) = -3q^2
\]
\[
q \cdot 6 = 6q
\]

Now, combine all these terms together:
\[
4p^3 + 2p^2 + p^2q - 12pq - 3q^2 + 24p + (-6q + 6q) + 12
\]

Simplifying, the terms involving \(q\) cancel out, so we are left with:
\[
4p^3 + 2p^2 + p^2q - 12pq - 3q^2 + 24p + 12
\]

This matches the third option in the image:
\[
\boxed{4p^3 + 2p^2 + p^2q - 12pq - 3q^2 + 24p + 12}
\]

Would you like more details on any step? Here are some additional questions to explore this further:

1. How would the expression change if one of the terms was squared?
2. Can this expression be factored after simplification?
3. What happens if we change the order of multiplication (commutativity)?
4. How would you approach this problem if it involved more variables?
5. What is the importance of grouping like terms in simplifying polynomial expressions?

**Tip:** Always keep track of positive and negative signs when distributing across multiple terms to avoid errors.

Simplify the following expression with distribution: (4p + 2 + q)(p^2 - 3q + 6)

Learn how to simplify the polynomial expression (4p + 2 + q)(p^2 - 3q + 6) using the distributive property. Step-by-step solution covering polynomial multiplication. Suitable for students in Grades 9-10.

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