Answer :
To determine which expressions are equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex], let's simplify each given expression step-by-step:
1. [tex]\(\left(-19x^2 - 4x - 7\right) + \left(4x^2 + 25x - 5\right)\)[/tex]
Combine like terms:
[tex]\[ -19x^2 + 4x^2 + (-4x + 25x) + (-7 - 5) = -15x^2 + 21x - 12 \][/tex]
This is not equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
2. [tex]\(\left(-17x^2 + 2x - 3\right) + \(\left(2x^2 - 31x - 9\right)\)[/tex]
Combine like terms:
[tex]\[ -17x^2 + 2x^2 + (2x - 31x) + (-3 - 9) = -15x^2 - 29x - 12 \][/tex]
This is equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
3. [tex]\(-2(7x + 1) - 5(3x^2 + 3x + 2)\)[/tex]
Simplify each term:
[tex]\[ -2(7x + 1) = -14x - 2 \][/tex]
[tex]\[ -5(3x^2 + 3x + 2) = -15x^2 - 15x - 10 \][/tex]
Combine like terms:
[tex]\[ -15x^2 - 14x - 15x - 2 - 10 = -15x^2 - 29x - 12 \][/tex]
This is equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
4. [tex]\(-2(4x - 15) - 3(5x^2 + 7x + 6)\)[/tex]
Simplify each term:
[tex]\[ -2(4x - 15) = -8x + 30 \][/tex]
[tex]\[ -3(5x^2 + 7x + 6) = -15x^2 - 21x - 18 \][/tex]
Combine like terms:
[tex]\[ -15x^2 - 8x - 21x + 30 - 18 = -15x^2 - 29x + 12 \][/tex]
This is not equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
5. [tex]\(\left(5x^2 - 10x + 8\right) - \left(10x^2 + 19x + 20\right)\)[/tex]
Distribute the negative sign:
[tex]\[ (5x^2 - 10x + 8) + (-10x^2 - 19x - 20) = 5x^2 - 10x^2 - 10x - 19x + 8 - 20 = -5x^2 - 29x - 12 \][/tex]
This is not equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
6. [tex]\(\left(-7x^2 - 21x + 13\right) - \left(8x^2 + 8x + 25\right)\)[/tex]
Distribute the negative sign:
[tex]\[ (-7x^2 - 21x + 13) + (-8x^2 - 8x - 25) = -7x^2 - 8x^2 - 21x - 8x + 13 - 25 = -15x^2 - 29x - 12 \][/tex]
This is equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
Therefore, the expressions that are equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex] are:
2, 3, and 6
1. [tex]\(\left(-19x^2 - 4x - 7\right) + \left(4x^2 + 25x - 5\right)\)[/tex]
Combine like terms:
[tex]\[ -19x^2 + 4x^2 + (-4x + 25x) + (-7 - 5) = -15x^2 + 21x - 12 \][/tex]
This is not equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
2. [tex]\(\left(-17x^2 + 2x - 3\right) + \(\left(2x^2 - 31x - 9\right)\)[/tex]
Combine like terms:
[tex]\[ -17x^2 + 2x^2 + (2x - 31x) + (-3 - 9) = -15x^2 - 29x - 12 \][/tex]
This is equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
3. [tex]\(-2(7x + 1) - 5(3x^2 + 3x + 2)\)[/tex]
Simplify each term:
[tex]\[ -2(7x + 1) = -14x - 2 \][/tex]
[tex]\[ -5(3x^2 + 3x + 2) = -15x^2 - 15x - 10 \][/tex]
Combine like terms:
[tex]\[ -15x^2 - 14x - 15x - 2 - 10 = -15x^2 - 29x - 12 \][/tex]
This is equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
4. [tex]\(-2(4x - 15) - 3(5x^2 + 7x + 6)\)[/tex]
Simplify each term:
[tex]\[ -2(4x - 15) = -8x + 30 \][/tex]
[tex]\[ -3(5x^2 + 7x + 6) = -15x^2 - 21x - 18 \][/tex]
Combine like terms:
[tex]\[ -15x^2 - 8x - 21x + 30 - 18 = -15x^2 - 29x + 12 \][/tex]
This is not equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
5. [tex]\(\left(5x^2 - 10x + 8\right) - \left(10x^2 + 19x + 20\right)\)[/tex]
Distribute the negative sign:
[tex]\[ (5x^2 - 10x + 8) + (-10x^2 - 19x - 20) = 5x^2 - 10x^2 - 10x - 19x + 8 - 20 = -5x^2 - 29x - 12 \][/tex]
This is not equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
6. [tex]\(\left(-7x^2 - 21x + 13\right) - \left(8x^2 + 8x + 25\right)\)[/tex]
Distribute the negative sign:
[tex]\[ (-7x^2 - 21x + 13) + (-8x^2 - 8x - 25) = -7x^2 - 8x^2 - 21x - 8x + 13 - 25 = -15x^2 - 29x - 12 \][/tex]
This is equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex].
Therefore, the expressions that are equivalent to [tex]\( -15x^2 - 29x - 12 \)[/tex] are:
2, 3, and 6
