To solve this problem and find the first term of the polynomial in standard form, let's simplify and organize the polynomial step-by-step.
Given polynomial:
[tex]\[ 4x^2 y^2 - 2y^4 - 8xy^3 + 9x^3 y + 6y^4 - 2xy^3 - 3x^4 + x^2 y^2 \][/tex]
### Step 1: Combine Like Terms
1. Combine the [tex]\( x^2 y^2 \)[/tex] terms:
[tex]\[ 4x^2 y^2 + x^2 y^2 = 5x^2 y^2 \][/tex]
2. Combine the [tex]\( y^4 \)[/tex] terms:
[tex]\[ -2y^4 + 6y^4 = 4y^4 \][/tex]
3. Combine the [tex]\( xy^3 \)[/tex] terms:
[tex]\[ -8xy^3 - 2xy^3 = -10xy^3 \][/tex]
4. The remaining terms ([tex]\(9x^3 y\)[/tex] and [tex]\(-3x^4\)[/tex]) don't have any like terms to combine with.
So, the combined polynomial is:
[tex]\[ 5x^2 y^2 + 4y^4 - 10xy^3 + 9x^3 y - 3x^4 \][/tex]
### Step 2: Write in Standard Form
In standard form, a polynomial is written with the highest degree terms first and in descending order of their degrees:
1. The term [tex]\(-3x^4\)[/tex] has the highest degree (degree 4).
2. The next highest degree terms are [tex]\(9x^3 y\)[/tex] and there are no repetitions.
3. The next term by degree is [tex]\(5x^2 y^2\)[/tex].
4. The term [tex]\(-10xy^3\)[/tex] follows by having degree 4 but arranged after the quadratic terms.
5. Lastly, [tex]\(4y^4\)[/tex].
Standard form then is:
[tex]\[ -3x^4 + 9x^3 y + 5x^2 y^2 - 10xy^3 + 4y^4 \][/tex]
### Conclusion
If the last term written by Julian is [tex]\(-3x^4\)[/tex] (which has the highest degree term - degree 4), then the first term of his polynomial in standard form must be the next highest degree term, which in our case is [tex]\(4y^4\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{4y^4} \][/tex]
