Let's analyze the given problem and the work done step-by-step.
We are multiplying the two polynomials [tex]\((3p - 7)\)[/tex] and [tex]\((2p^2 - 3p - 4)\)[/tex] using the table provided.
[tex]\[
(3p - 7) \left( 2p^2 - 3p - 4 \right)
\][/tex]
The table given looks like:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& 2p^2 & -3p & -4 \\
\hline
3p & 6p^3 & -9p^2 & -12p \\
\hline
-7 & -14p^2 & 21p & 28 \\
\hline
\end{array}
\][/tex]
Let's interpret what this table represents:
1. Multiplying [tex]\(3p\)[/tex] with each term of the second polynomial:
[tex]\[
3p \cdot 2p^2 = 6p^3
\][/tex]
[tex]\[
3p \cdot (-3p) = -9p^2
\][/tex]
[tex]\[
3p \cdot (-4) = -12p
\][/tex]
2. Multiplying [tex]\(-7\)[/tex] with each term of the second polynomial:
[tex]\[
-7 \cdot 2p^2 = -14p^2
\][/tex]
[tex]\[
-7 \cdot (-3p) = 21p
\][/tex]
[tex]\[
-7 \cdot (-4) = 28
\][/tex]
Now, combining the terms according to their degrees:
- [tex]\(p^3\)[/tex] term:
[tex]\[
6p^3
\][/tex]
- [tex]\(p^2\)[/tex] terms:
[tex]\[
-9p^2 + (-14p^2) = -23p^2
\][/tex]
- [tex]\(p\)[/tex] terms:
[tex]\[
-12p + 21p = 9p
\][/tex]
- Constant term:
[tex]\[
28
\][/tex]
Thus, combining all these results, the final polynomial is:
[tex]\[
6p^3 - 23p^2 + 9p + 28
\][/tex]
So, the correct product is:
[tex]\[
6p^3 - 23p^2 + 9p + 28
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{6 p^3 - 23 p^2 + 9 p + 28}
\][/tex]
