To determine which polynomials are listed with their correct additive inverse, we need to check if the sum of each polynomial and its supposed additive inverse equals zero. Recall that the additive inverse of a polynomial [tex]\( P(x) \)[/tex] is a polynomial [tex]\( -P(x) \)[/tex] such that [tex]\( P(x) + (-P(x)) = 0 \)[/tex].
1. Polynomial: [tex]\( x^2 + 3x - 2 \)[/tex], Inverse: [tex]\( -x^2 - 3x + 2 \)[/tex]
[tex]\[
(x^2 + 3x - 2) + (-x^2 - 3x + 2) = x^2 - x^2 + 3x - 3x - 2 + 2 = 0
\][/tex]
This is a correct additive inverse.
2. Polynomial: [tex]\( -y^7 - 10 \)[/tex], Inverse: [tex]\( -y^7 + 10 \)[/tex]
[tex]\[
(-y^7 - 10) + (-y^7 + 10) = -y^7 - y^7 - 10 + 10 = -2y^7 \neq 0
\][/tex]
This is not a correct additive inverse.
3. Polynomial: [tex]\( 6z^5 + 6z^5 - 6z^4 \)[/tex], Inverse: [tex]\( -6z^5 - 6z^5 + 6z^4 \)[/tex]
[tex]\[
(6z^5 + 6z^5 - 6z^4) + (-6z^5 - 6z^5 + 6z^4) = 6z^5 - 6z^5 + 6z^5 - 6z^5 - 6z^4 + 6z^4 = 0
\][/tex]
This is a correct additive inverse.
4. Polynomial: [tex]\( x - 1 \)[/tex], Inverse: [tex]\( 1 - x \)[/tex]
[tex]\[
(x - 1) + (1 - x) = x - x - 1 + 1 = 0
\][/tex]
This is a correct additive inverse.
5. Polynomial: [tex]\( -5x^2 - 2x - 10 \)[/tex], Inverse: [tex]\( 5x^2 - 2x + 10 \)[/tex]
[tex]\[
(-5x^2 - 2x - 10) + (5x^2 - 2x + 10) = -5x^2 + 5x^2 - 2x - 2x - 10 + 10 = -4x \neq 0
\][/tex]
This is not a correct additive inverse.
Given these calculations, the polynomials listed with their correct additive inverses are:
[tex]\[ \boxed{1, 3, 4} \][/tex]
