To solve this problem, we need to combine like terms and express the polynomial in standard form. Let's start by identifying and combining the like terms in the polynomial.
The given polynomial is:
[tex]\[ 9 x y^3 - 4 y^4 - 10 x^2 y^2 + x^3 y + 3 x^4 + 2 x^2 y^2 - 9 y^4 \][/tex]
First, let's group the like terms together:
1. [tex]\(x^4\)[/tex] term:
[tex]\[ 3 x^4 \][/tex]
2. [tex]\(x^3 y\)[/tex] term:
[tex]\[ x^3 y \][/tex]
3. [tex]\(x^2 y^2\)[/tex] terms:
[tex]\[ -10 x^2 y^2 + 2 x^2 y^2 = -8 x^2 y^2 \][/tex]
4. [tex]\(x y^3\)[/tex] term:
[tex]\[ 9 x y^3 \][/tex]
5. [tex]\(y^4\)[/tex] terms:
[tex]\[ -4 y^4 - 9 y^4 = -13 y^4 \][/tex]
Now that we have combined the like terms, we can write the polynomial in standard form by ordering the terms from the highest degree to the lowest degree:
[tex]\[ 3 x^4 + x^3 y - 8 x^2 y^2 + 9 x y^3 - 13 y^4 \][/tex]
Therefore, the polynomial in standard form is:
[tex]\[ \boxed{3 x^4 + x^3 y - 8 x^2 y^2 + 9 x y^3 - 13 y^4} \][/tex]
