Explore IDNLearn.com's extensive Q&A database and find the answers you're looking for. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.

What is the product of [tex]2p + q[/tex] and [tex]-3q - 6p + 1[/tex]?

A. [tex]-12p^2 - 6pq - 4p - 3q + 1[/tex]
B. [tex]-12p^2 - 12pq + 2p - 3q^2 + q[/tex]
C. [tex]-9p^2q^2 + 12pq - 2p + q[/tex]
D. [tex]12p^2 + 12pq + 2p + 3q^2 + q[/tex]


Sagot :

To find the product of the expressions [tex]\(2p + q\)[/tex] and [tex]\(-3q - 6p + 1\)[/tex], follow these steps:

1. Write out the given expressions:
[tex]\[ (2p + q)(-3q - 6p + 1) \][/tex]

2. Distribute each term in the first expression to each term in the second expression:
[tex]\[ (2p)(-3q) + (2p)(-6p) + (2p)(1) + (q)(-3q) + (q)(-6p) + (q)(1) \][/tex]

Evaluate each term:
- [tex]\( (2p)(-3q) = -6pq \)[/tex]
- [tex]\( (2p)(-6p) = -12p^2 \)[/tex]
- [tex]\( (2p)(1) = 2p \)[/tex]
- [tex]\( (q)(-3q) = -3q^2 \)[/tex]
- [tex]\( (q)(-6p) = -6pq \)[/tex]
- [tex]\( (q)(1) = q \)[/tex]

3. Combine all the terms together:
[tex]\[ -6pq - 12p^2 + 2p - 3q^2 - 6pq + q \][/tex]

4. Combine like terms:
- Combine the [tex]\(pq\)[/tex] terms: [tex]\(-6pq - 6pq = -12pq\)[/tex]
- The final expression after combining like terms is:
[tex]\[ -12p^2 - 12pq + 2p - 3q^2 + q \][/tex]

5. Compare with the given options:

The product we obtained is:
[tex]\[ -12p^2 - 12pq + 2p - 3q^2 + q \][/tex]

Match this with the given options:

- Option [tex]\(\mathbf{1}: -12p^2 - 6pq - 4p - 3q + 1\)[/tex]
- Option [tex]\(\mathbf{2}: -12p^2 - 12pq + 2p - 3q^2 + q\)[/tex]
- Option [tex]\(\mathbf{3}: -9p^2q^2 + 12pq - 2p + q\)[/tex]
- Option [tex]\(\mathbf{4}: 12p^2 + 12pq + 2p + 3q^2 + q\)[/tex]

The correct match is:
[tex]\[ \textbf{Option 2: } -12p^2 - 12pq + 2p - 3q^2 + q \][/tex]

Therefore, the product of [tex]\(2p + q\)[/tex] and [tex]\(-3q - 6p + 1\)[/tex] is:
[tex]\[ -12p^2 - 12pq + 2p - 3q^2 + q \][/tex]

So, the correct option is [tex]\(\boxed{2}\)[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.