Certainly! Let's find the difference of the given polynomials step by step.
Given polynomials:
[tex]\[ 3n^2(n^2 + 4n - 5) - (2n^2 - n^4 + 3) \][/tex]
### Step 1: Expand the first polynomial
[tex]\[ 3n^2(n^2 + 4n - 5) \][/tex]
Distribute [tex]\(3n^2\)[/tex] across each term inside the parentheses:
[tex]\[ 3n^2 \cdot n^2 + 3n^2 \cdot 4n - 3n^2 \cdot 5 \][/tex]
[tex]\[ = 3n^4 + 12n^3 - 15n^2 \][/tex]
So, the expanded form of the first polynomial is:
[tex]\[ 3n^4 + 12n^3 - 15n^2 \][/tex]
### Step 2: Write the second polynomial in its simplified form
The second polynomial is:
[tex]\[ 2n^2 - n^4 + 3 \][/tex]
### Step 3: Subtract the second polynomial from the expanded first polynomial
[tex]\[ (3n^4 + 12n^3 - 15n^2) - (2n^2 - n^4 + 3) \][/tex]
First, distribute the negative sign across each term in the second polynomial:
[tex]\[ 3n^4 + 12n^3 - 15n^2 - 2n^2 + n^4 - 3 \][/tex]
Next, combine like terms:
[tex]\[ 3n^4 + n^4 + 12n^3 - 15n^2 - 2n^2 - 3 \][/tex]
Simplify the polynomial:
[tex]\[ = (3n^4 + n^4) + 12n^3 + (-15n^2 - 2n^2) - 3 \][/tex]
[tex]\[ = 4n^4 + 12n^3 - 17n^2 - 3 \][/tex]
### Step 4: Determine the degree and number of terms
The simplified polynomial is:
[tex]\[ 4n^4 + 12n^3 - 17n^2 - 3 \][/tex]
- Degree: The highest power of [tex]\(n\)[/tex] is 4, so the degree is 4.
- Number of Terms: There are four distinct terms: [tex]\(4n^4\)[/tex], [tex]\(12n^3\)[/tex], [tex]\(-17n^2\)[/tex], and [tex]\(-3\)[/tex].
Therefore, the resultant polynomial is a [tex]\(4^{\text{th}}\)[/tex]-degree polynomial with 4 terms.
### Answer
[tex]\[ \text{Final classification:} \, \boxed{\text{B. } 4^{\text {th }} \text{ degree polynomial with 4 terms}.} \][/tex]
