To solve the given problem, we need to follow these steps:
1. Expand the first polynomial [tex]\( 3n^2(n^2 + 4n - 5) \)[/tex].
2. Simplify the subtraction between the expanded polynomial and the second polynomial [tex]\( 2n^2 - n^4 + 3 \)[/tex].
3. Determine the degree of the resulting polynomial.
4. Count the number of terms in the simplified polynomial.
Let's go through these steps one by one.
### Step 1: Expand the first polynomial
The first polynomial is:
[tex]\[ 3n^2(n^2 + 4n - 5) \][/tex]
Distribute [tex]\( 3n^2 \)[/tex] inside the parenthesis:
[tex]\[ 3n^2 \cdot n^2 + 3n^2 \cdot 4n + 3n^2 \cdot (-5) \][/tex]
[tex]\[ 3n^2 \cdot n^2 = 3n^4 \][/tex]
[tex]\[ 3n^2 \cdot 4n = 12n^3 \][/tex]
[tex]\[ 3n^2 \cdot (-5) = -15n^2 \][/tex]
Combining these, we get:
[tex]\[ 3n^4 + 12n^3 - 15n^2 \][/tex]
### Step 2: Subtract the second polynomial
The second polynomial is:
[tex]\[ 2n^2 - n^4 + 3 \][/tex]
We need to subtract this from the expanded first polynomial:
[tex]\[ (3n^4 + 12n^3 - 15n^2) - (2n^2 - n^4 + 3) \][/tex]
Perform the subtraction term by term:
[tex]\[ 3n^4 + 12n^3 - 15n^2 - 2n^2 + n^4 - 3 \][/tex]
Combine like terms:
[tex]\[ 3n^4 + n^4 + 12n^3 - 15n^2 - 2n^2 - 3 \][/tex]
[tex]\[ 4n^4 + 12n^3 - 17n^2 - 3 \][/tex]
### Step 3: Determine the degree of the polynomial
The degree of a polynomial is the highest power of the variable [tex]\( n \)[/tex] in the polynomial. In:
[tex]\[ 4n^4 + 12n^3 - 17n^2 - 3 \][/tex]
The highest power of [tex]\( n \)[/tex] is 4. Therefore, this polynomial is of degree 4.
### Step 4: Count the number of terms
The number of terms in the polynomial [tex]\( 4n^4 + 12n^3 - 17n^2 - 3 \)[/tex] is 4.
### Conclusion
The simplified polynomial is [tex]\( 4n^4 + 12n^3 - 17n^2 - 3 \)[/tex]. It is a 4th-degree polynomial with 4 terms.
Thus, the correct answer is:
[tex]\[ \boxed{D. 4^{\text{th}} \text{ degree polynomial with 4 terms}} \][/tex]
