Let's analyze each expression to see which of them are equivalent to the polynomial [tex]\(-15x^2 - 29x - 12\)[/tex].
### Expression 1
[tex]\[
\left(-7x^2 - 21x + 13\right) - \left(8x^2 + 8x + 25\right)
\][/tex]
Combine and simplify:
[tex]\[
= (-7x^2 - 21x + 13) - 8x^2 - 8x - 25
\][/tex]
[tex]\[
= -7x^2 - 21x + 13 - 8x^2 - 8x - 25
\][/tex]
[tex]\[
= -15x^2 - 29x - 12
\][/tex]
The simplified expression matches the given polynomial. Thus, Expression 1 is correct.
### Expression 2
[tex]\[
-2(4x - 15) - 3(5x^2 + 7x + 6)
\][/tex]
Distribute and simplify:
[tex]\[
= -2(4x) + 30 - 3(5x^2) - 3(7x) - 3(6)
\][/tex]
[tex]\[
= -8x + 30 - 15x^2 - 21x - 18
\][/tex]
[tex]\[
= -15x^2 - 29x + 12
\][/tex]
The simplified form does not match the given polynomial. Thus, Expression 2 is not correct.
### Expression 3
[tex]\[
\left(-19x^2 - 4x - 7\right) + \left(4x^2 + 25x - 5\right)
\][/tex]
Combine and simplify:
[tex]\[
= -19x^2 - 4x - 7 + 4x^2 + 25x - 5
\][/tex]
[tex]\[
= -15x^2 + 21x - 12
\][/tex]
The simplified form does not match the given polynomial. Thus, Expression 3 is not correct.
### Expression 4
[tex]\[
\left(-17x^2 + 2x - 3\right) + \left(2x^2 - 31x - 9\right)
\][/tex]
Combine and simplify:
[tex]\[
= -17x^2 + 2x - 3 + 2x^2 - 31x - 9
\][/tex]
[tex]\[
= -15x^2 - 29x - 12
\][/tex]
The simplified expression matches the given polynomial. Thus, Expression 4 is correct.
### Expression 5
[tex]\[
-2(7x + 1) - 5(3x^2 + 3x + 2)
\][/tex]
Distribute and simplify:
[tex]\[
= -2(7x) - 2(1) - 5(3x^2) - 5(3x) - 5(2)
\][/tex]
[tex]\[
= -14x - 2 - 15x^2 - 15x - 10
\][/tex]
[tex]\[
= -15x^2 - 29x - 12
\][/tex]
The simplified expression matches the given polynomial. Thus, Expression 5 is correct.
### Expression 6
[tex]\[
\left(5x^2 - 10x + 8\right) - \left(10x^2 + 19x + 20\right)
\][/tex]
Combine and simplify:
[tex]\[
= 5x^2 - 10x + 8 - 10x^2 - 19x - 20
\][/tex]
[tex]\[
= -5x^2 - 29x - 12
\][/tex]
The simplified form does not match the given polynomial. Thus, Expression 6 is not correct.
### Conclusion
The expressions equivalent to the given polynomial [tex]\(-15x^2 - 29x - 12\)[/tex] are:
[tex]\[ \boxed{1, 4, 5} \][/tex]
