To simplify the polynomial expression [tex]\((7n+1)(3n+5)-(4n-2)(3n+1)\)[/tex], we will follow these steps:
1. Expand each polynomial expression separately:
- Expand [tex]\((7n + 1)(3n + 5)\)[/tex]
- Expand [tex]\((4n - 2)(3n + 1)\)[/tex]
2. Combine the expanded expressions and simplify.
Let's start with expanding [tex]\((7n + 1)(3n + 5)\)[/tex]:
[tex]\[
(7n + 1)(3n + 5)
\][/tex]
Using the distributive property (FOIL), we expand this:
[tex]\[
7n \cdot 3n + 7n \cdot 5 + 1 \cdot 3n + 1 \cdot 5
\][/tex]
This results in:
[tex]\[
21n^2 + 35n + 3n + 5
\][/tex]
Combining like terms:
[tex]\[
21n^2 + 38n + 5
\][/tex]
Next, let's expand [tex]\((4n - 2)(3n + 1)\)[/tex]:
[tex]\[
(4n - 2)(3n + 1)
\][/tex]
Using the distributive property (FOIL), we expand this:
[tex]\[
4n \cdot 3n + 4n \cdot 1 - 2 \cdot 3n - 2 \cdot 1
\][/tex]
This results in:
[tex]\[
12n^2 + 4n - 6n - 2
\][/tex]
Combining like terms:
[tex]\[
12n^2 - 2n - 2
\][/tex]
Now subtract the second expanded expression from the first expanded expression:
[tex]\[
(21n^2 + 38n + 5) - (12n^2 - 2n - 2)
\][/tex]
Distribute the negative sign:
[tex]\[
21n^2 + 38n + 5 - 12n^2 + 2n + 2
\][/tex]
Combine the like terms:
[tex]\[
(21n^2 - 12n^2) + (38n + 2n) + (5 + 2)
\][/tex]
Simplify each term:
[tex]\[
9n^2 + 40n + 7
\][/tex]
Thus, the simplified polynomial expression is:
[tex]\[
\boxed{9n^2 + 40n + 7}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{A. \, 9n^2 + 40n + 7}
\][/tex]
