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].
