To solve the expression [tex]\(\left(4 x^2 y^3 + 2 x y^2 - 2 y\right) - \left(-7 x^2 y^3 + 6 x y^2 - 2 y\right)\)[/tex], we need to follow these steps:
1. Distribute the negative sign through the second polynomial:
[tex]\[
- \left(-7 x^2 y^3 + 6 x y^2 - 2 y\right) = 7 x^2 y^3 - 6 x y^2 + 2 y
\][/tex]
2. Combine like terms from the two polynomials:
[tex]\[
(4 x^2 y^3 + 2 x y^2 - 2 y) + (7 x^2 y^3 - 6 x y^2 + 2 y)
\][/tex]
3. Add the coefficients of [tex]\(x^2 y^3\)[/tex]:
[tex]\[
4 x^2 y^3 + 7 x^2 y^3 = (4 + 7)x^2 y^3 = 11 x^2 y^3
\][/tex]
4. Add the coefficients of [tex]\(x y^2\)[/tex]:
[tex]\[
2 x y^2 - 6 x y^2 = (2 - 6)x y^2 = -4 x y^2
\][/tex]
5. Add the coefficients of [tex]\(y\)[/tex]:
[tex]\[
-2 y + 2 y = (-2 + 2)y = 0 y
\][/tex]
6. Combine the results:
[tex]\[
11 x^2 y^3 - 4 x y^2 + 0 y
\][/tex]
Thus, the coefficients in the difference are:
[tex]\[ 11 \quad x^2 y^3 \quad + \quad (-4) \quad x y^2 \quad + \quad 0 \quad y \][/tex]
So, the final answer is:
[tex]\[ 11 \ x^2 y^3 + (-4) \ x y^2 + 0 \ y \][/tex]
Or more simply:
[tex]\[ 11 x^2 y^3 - 4 x y^2 \][/tex]
The correct coefficients for the difference are:
[tex]\[
\boxed{11} \ x^2 y^3 + \boxed{-4} \ x y^2 + \boxed{0} \ y
\][/tex]
