To solve the problem of finding the difference between the two given polynomials and classifying the resulting polynomial in terms of its degree and number of terms, let's proceed step-by-step:
1. Expand and Simplify the Given Polynomials:
The first polynomial is:
[tex]\[
3n^2 \left(n^2 + 4n - 5\right)
\][/tex]
Let's expand this polynomial.
[tex]\[
3n^2 \cdot n^2 + 3n^2 \cdot 4n - 3n^2 \cdot 5 = 3n^4 + 12n^3 - 15n^2
\][/tex]
The second polynomial is:
[tex]\[
2n^2 - n^4 + 3
\][/tex]
2. Subtract the Second Polynomial from the First Polynomial:
[tex]\[
\left(3n^4 + 12n^3 - 15n^2\right) - \left(2n^2 - n^4 + 3\right)
\][/tex]
Distribute the negative sign through the second polynomial:
[tex]\[
3n^4 + 12n^3 - 15n^2 - 2n^2 + n^4 - 3
\][/tex]
Combine like terms:
[tex]\[
(3n^4 + n^4) + 12n^3 + (-15n^2 - 2n^2) - 3 = 4n^4 + 12n^3 - 17n^2 - 3
\][/tex]
This results in:
[tex]\[
4n^4 + 12n^3 - 17n^2 - 3
\][/tex]
3. Determine the Degree of the Resulting Polynomial:
The degree of a polynomial is the highest power of the variable [tex]\(n\)[/tex] in it. In this case, the highest exponent of [tex]\(n\)[/tex] is 4.
4. Count the Number of Terms in the Resulting Polynomial:
The resulting polynomial [tex]\(4n^4 + 12n^3 - 17n^2 - 3\)[/tex] has 4 distinct terms:
[tex]\[
4n^4, \ 12n^3, \ -17n^2, \text{ and } -3
\][/tex]
5. Classify the Polynomial:
The polynomial we obtained is a 4th-degree polynomial with 4 terms.
Therefore, the correct classification of the polynomial difference is:
[tex]\[
\boxed{\text{C. 4th degree polynomial with 4 terms}}
\][/tex]
