Let's start by fully simplifying the given polynomial by combining like terms:
The polynomial is:
[tex]\[ 4x^2y^2 - 2y^4 - 8xy^3 + 9x^3y + 6y^4 - 2xy^3 - 3x^4 + x^2y^2 \][/tex]
First, we group like terms together:
### Grouping the like terms:
1. [tex]\(x^4\)[/tex] terms:
[tex]\[
-3x^4
\][/tex]
2. [tex]\(x^3y\)[/tex] terms:
[tex]\[
9x^3y
\][/tex]
3. [tex]\(x^2y^2\)[/tex] terms:
[tex]\[
4x^2y^2 + x^2y^2 = 5x^2y^2
\][/tex]
4. [tex]\(xy^3\)[/tex] terms:
[tex]\[
-8xy^3 - 2xy^3 = -10xy^3
\][/tex]
5. [tex]\(y^4\)[/tex] terms:
[tex]\[
-2y^4 + 6y^4 = 4y^4
\][/tex]
So, the simplified polynomial is:
[tex]\[
-3x^4 + 9x^3y + 5x^2y^2 - 10xy^3 + 4y^4
\][/tex]
Julian wrote the last term as [tex]\( -3x^4 \)[/tex]. In standard form, the polynomial is arranged in descending order of the powers of x and y. Therefore, the first term should be the one with the highest power of y, which is:
[tex]\[ 4y^4 \][/tex]
Thus, the correct answer is:
[tex]\[ 4y^4 \][/tex]
