To find the difference of the two polynomials [tex]\((-2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4) - (6 x^4 y - 5 x^2 y^3 - y^5)\)[/tex], we must subtract each term in the second polynomial from the corresponding terms in the first polynomial.
Given polynomials are:
[tex]\[ P_1 = -2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4 \][/tex]
[tex]\[ P_2 = 6 x^4 y - 5 x^2 y^3 - y^5 \][/tex]
Now we perform the subtraction [tex]\( P_1 - P_2 \)[/tex]:
[tex]\[ (-2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4) - (6 x^4 y - 5 x^2 y^3 - y^5) \][/tex]
Distribute the subtraction across the terms in [tex]\( P_2 \)[/tex]:
[tex]\[ = -2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4 - 6 x^4 y + 5 x^2 y^3 + y^5 \][/tex]
Now, combine like terms:
1. There is only one [tex]\( x^4 y \)[/tex] term: [tex]\(-6 x^4 y \)[/tex].
2. There is only one [tex]\( x^3 y^2 \)[/tex] term: [tex]\( -2 x^3 y^2 \)[/tex].
3. Combine [tex]\( x^2 y^3 \)[/tex] terms: [tex]\( 4 x^2 y^3 + 5 x^2 y^3 = 9 x^2 y^3 \)[/tex].
4. There is only one [tex]\( x y^4 \)[/tex] term: [tex]\( -3 x y^4 \)[/tex].
5. There is only one [tex]\( y^5 \)[/tex] term: [tex]\( + y^5 \)[/tex].
Putting these together, we get:
[tex]\[ -6 x^4 y - 2 x^3 y^2 + 9 x^2 y^3 - 3 x y^4 + y^5 \][/tex]
So, the result is:
[tex]\[ -6 x^4 y - 2 x^3 y^2 + 9 x^2 y^3 - 3 x y^4 + y^5 \][/tex]
Among the given choices, this matches with:
[tex]\[ -6 x^4 y - 2 x^3 y^2 + 9 x^2 y^3 - 3 x y^4 + y^5 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-6 x^4 y-2 x^3 y^2+9 x^2 y^3-3 x y^4+y^5} \][/tex]
