To match each expression to its factored form, we carefully analyze each polynomial in its expanded form and associate it with the appropriate match from the given choices. Here's the step-by-step reasoning:
1. Expression a:
[tex]\[
2 x\left(3 x^3-2 x+7\right)
\][/tex]
By distributing [tex]\(2x\)[/tex] to the terms inside the parentheses:
[tex]\[
2x \cdot 3x^3 - 2x \cdot 2x + 2x \cdot 7 = 6x^4 - 4x^2 + 14x
\][/tex]
Thus, it matches:
[tex]\[
6 x^4-4 x^2+14 x \quad \text{(Number 3)}
\][/tex]
Therefore, [tex]\( a \rightarrow 3 \)[/tex].
2. Expression b:
[tex]\[
7 x^2(8 x-1)
\][/tex]
By distributing [tex]\(7x^2\)[/tex] to the terms inside the parentheses:
[tex]\[
7x^2 \cdot 8x - 7x^2 \cdot 1 = 56x^3 - 7x^2
\][/tex]
Thus, it matches:
[tex]\[
56 x^3-7 x^2 \quad \text{(Number 1)}
\][/tex]
Therefore, [tex]\( b \rightarrow 1 \)[/tex].
3. Expression c:
[tex]\[
9\left(2 x^2+3 x-1\right)
\][/tex]
By distributing [tex]\(9\)[/tex] to the terms inside the parentheses:
[tex]\[
9 \cdot 2x^2 + 9 \cdot 3x - 9 \cdot 1 = 18x^2 + 27x - 9
\][/tex]
Thus, it matches:
[tex]\[
18 x^2 + 27 x - 9 \quad \text{(Number 4)}
\][/tex]
Therefore, [tex]\( c \rightarrow 4 \)[/tex].
4. Expression d:
[tex]\[
4\left(x^2+8\right)
\][/tex]
By distributing [tex]\(4\)[/tex] to the terms inside the parentheses:
[tex]\[
4 \cdot x^2 + 4 \cdot 8 = 4x^2 + 32
\][/tex]
Thus, it matches:
[tex]\[
4 x^2 + 32 \quad \text{(Number 2)}
\][/tex]
Therefore, [tex]\( d \rightarrow 2 \)[/tex].
Summarizing the matches:
- [tex]\( a \rightarrow 3 \)[/tex]
- [tex]\( b \rightarrow 1 \)[/tex]
- [tex]\( c \rightarrow 4 \)[/tex]
- [tex]\( d \rightarrow 2 \)[/tex]
Therefore, the final matching is:
[tex]\[
\{a: 3, b: 1, c: 4, d: 2\}
\][/tex]
