Answer :
To match each polynomial expression on the left with an equivalent expression on the right, we'll evaluate each polynomial step by step:
1. Expression on the left:
[tex]\[
3 x\left(1 + 16 x + 2 x^2\right)
\][/tex]
Equivalent expression on the right:
[tex]\[
3\left(2 x^2 + 1\right)(x + 8)
\][/tex]
2. Expression on the left:
[tex]\[
-24 + 48 x^2 - 3 x + 6 x^3
\][/tex]
Equivalent expression on the right:
[tex]\[
-24 + 48 x^2 - 3 x + 6 x^3
\][/tex]
3. Expression on the left:
[tex]\[
\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)
\][/tex]
Equivalent expression on the right:
[tex]\[
\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)
\][/tex]
4. Expression on the left:
[tex]\[
-4 x + 6 x^3 - 24 + 48 x^2
\][/tex]
Equivalent expression on the right:
[tex]\[
-4 x + 6 x^3 - 24 + 48 x^2
\][/tex]
5. Expression on the left:
[tex]\[
3\left(2 x^2 + 1\right)(x + 8)
\][/tex]
Equivalent expression on the right:
[tex]\[
3 x\left(1 + 16 x + 2 x^2\right)
\][/tex]
6. Expression on the left:
[tex]\[
\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)
\][/tex]
Equivalent expression on the right:
[tex]\[
\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)
\][/tex]
Putting it all together, the matches are as follows:
1. [tex]\(3 x\left(1 + 16 x + 2 x^2\right)\)[/tex] matches with [tex]\(3\left(2 x^2 + 1\right)(x + 8)\)[/tex].
2. [tex]\(-24 + 48 x^2 - 3 x + 6 x^3\)[/tex] matches with [tex]\(-24 + 48 x^2 - 3 x + 6 x^3\)[/tex].
3. [tex]\(\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)\)[/tex] matches with [tex]\(\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)\)[/tex].
4. [tex]\(-4 x + 6 x^3 - 24 + 48 x^2\)[/tex] matches with [tex]\(-4 x + 6 x^3 - 24 + 48 x^2\)[/tex].
5. [tex]\(3\left(2 x^2 + 1\right)(x + 8)\)[/tex] matches with [tex]\(3 x\left(1 + 16 x + 2 x^2\right)\)[/tex].
6. [tex]\(\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)\)[/tex] matches with [tex]\(\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)\)[/tex].
1. Expression on the left:
[tex]\[
3 x\left(1 + 16 x + 2 x^2\right)
\][/tex]
Equivalent expression on the right:
[tex]\[
3\left(2 x^2 + 1\right)(x + 8)
\][/tex]
2. Expression on the left:
[tex]\[
-24 + 48 x^2 - 3 x + 6 x^3
\][/tex]
Equivalent expression on the right:
[tex]\[
-24 + 48 x^2 - 3 x + 6 x^3
\][/tex]
3. Expression on the left:
[tex]\[
\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)
\][/tex]
Equivalent expression on the right:
[tex]\[
\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)
\][/tex]
4. Expression on the left:
[tex]\[
-4 x + 6 x^3 - 24 + 48 x^2
\][/tex]
Equivalent expression on the right:
[tex]\[
-4 x + 6 x^3 - 24 + 48 x^2
\][/tex]
5. Expression on the left:
[tex]\[
3\left(2 x^2 + 1\right)(x + 8)
\][/tex]
Equivalent expression on the right:
[tex]\[
3 x\left(1 + 16 x + 2 x^2\right)
\][/tex]
6. Expression on the left:
[tex]\[
\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)
\][/tex]
Equivalent expression on the right:
[tex]\[
\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)
\][/tex]
Putting it all together, the matches are as follows:
1. [tex]\(3 x\left(1 + 16 x + 2 x^2\right)\)[/tex] matches with [tex]\(3\left(2 x^2 + 1\right)(x + 8)\)[/tex].
2. [tex]\(-24 + 48 x^2 - 3 x + 6 x^3\)[/tex] matches with [tex]\(-24 + 48 x^2 - 3 x + 6 x^3\)[/tex].
3. [tex]\(\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)\)[/tex] matches with [tex]\(\left(4 x^3 + 48 x^2 + 3 x\right) + \left(2 x^3 + x + 24\right)\)[/tex].
4. [tex]\(-4 x + 6 x^3 - 24 + 48 x^2\)[/tex] matches with [tex]\(-4 x + 6 x^3 - 24 + 48 x^2\)[/tex].
5. [tex]\(3\left(2 x^2 + 1\right)(x + 8)\)[/tex] matches with [tex]\(3 x\left(1 + 16 x + 2 x^2\right)\)[/tex].
6. [tex]\(\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)\)[/tex] matches with [tex]\(\left(7 x^3 + 48 x^2 - 24\right) - \left(x^3 - 4 x - 24\right)\)[/tex].
