Let's analyze and solve the given problem step by step.
We start with two polynomial expressions:
[tex]\[ 3n^2(5n^2 - 2n + 1) + (4n^2 - 11n^4 - 9) \][/tex]
First, we need to expand and simplify each polynomial expression individually.
### Expanding the First Polynomial
[tex]\[ 3n^2(5n^2 - 2n + 1) \][/tex]
Distribute [tex]\(3n^2\)[/tex] across each term inside the parentheses:
[tex]\[
= 3n^2 \cdot 5n^2 - 3n^2 \cdot 2n + 3n^2 \cdot 1
= 15n^4 - 6n^3 + 3n^2
\][/tex]
So, the first polynomial simplifies to:
[tex]\[ 15n^4 - 6n^3 + 3n^2 \][/tex]
### Simplifying the Second Polynomial
The second polynomial expression is:
[tex]\[ 4n^2 - 11n^4 - 9 \][/tex]
This polynomial is already expanded, so it remains as:
[tex]\[ 4n^2 - 11n^4 - 9 \][/tex]
### Summing the Two Polynomials
Now, we add the two expanded polynomials together:
[tex]\[ (15n^4 - 6n^3 + 3n^2) + (4n^2 - 11n^4 - 9) \][/tex]
Combine like terms:
[tex]\[
15n^4 - 11n^4 + (-6n^3) + 3n^2 + 4n^2 - 9
= 4n^4 - 6n^3 + 7n^2 - 9
\][/tex]
The sum of the polynomials is:
[tex]\[ 4n^4 - 6n^3 + 7n^2 - 9 \][/tex]
### Classification of the Polynomial
1. Degree of the Polynomial: The degree is the highest power of [tex]\( n \)[/tex], which in this case is [tex]\( 4 \)[/tex]. So, it is a 4th degree polynomial.
2. Number of Terms: The polynomial [tex]\( 4n^4 - 6n^3 + 7n^2 - 9 \)[/tex] has 4 terms (each term separated by a plus or minus sign).
Therefore, the polynomial is a 4th degree polynomial with 4 terms.
So, the correct answer is:
[tex]\[ \boxed{\text{C. } 4^{\text{th}} \text{ degree polynomial with 4 terms}} \][/tex]
