Answer :
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]
### 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]
