To determine which expressions are equivalent to the polynomial [tex]\( -15x^2 - 29x - 12 \)[/tex], we'll expand and simplify each expression step-by-step.
1. Expression 1: [tex]\( -2(7x + 1) - 5(3x^2 + 3x + 2) \)[/tex]
Expand and simplify:
[tex]\[
\begin{align*}
-2(7x + 1) &= -14x - 2, \\
-5(3x^2 + 3x + 2) &= -15x^2 - 15x - 10, \\
-14x - 2 - 15x^2 - 15x - 10 &= -15x^2 - 29x - 12.
\end{align*}
\][/tex]
This matches the target polynomial.
2. Expression 2: [tex]\( (-17x^2 + 2x - 3) + (2x^2 - 31x - 9) \)[/tex]
Expand and simplify:
[tex]\[
\begin{align*}
-17x^2 + 2x - 3 + 2x^2 - 31x - 9 &= (-17x^2 + 2x^2) + (2x - 31x) + (-3 - 9) \\
&= -15x^2 - 29x - 12.
\end{align*}
\][/tex]
This matches the target polynomial.
3. Expression 3: [tex]\( (-7x^2 - 21x + 13) - (8x^2 + 8x + 25) \)[/tex]
Expand and simplify:
[tex]\[
\begin{align*}
-7x^2 - 21x + 13 - 8x^2 - 8x - 25 &= (-7x^2 - 8x^2) + (-21x - 8x) + (13 - 25) \\
&= -15x^2 - 29x - 12.
\end{align*}
\][/tex]
This matches the target polynomial.
4. Expression 4: [tex]\( (5x^2 - 10x + 8) - (10x^2 + 19x + 20) \)[/tex]
Expand and simplify:
[tex]\[
\begin{align*}
5x^2 - 10x + 8 - 10x^2 - 19x - 20 &= (5x^2 - 10x^2) + (-10x - 19x) + (8 - 20) \\
&= -5x^2 - 29x - 12.
\end{align*}
\][/tex]
This does not match the target polynomial.
5. Expression 5: [tex]\( -2(4x - 15) - 3(5x^2 + 7x + 6) \)[/tex]
Expand and simplify:
[tex]\[
\begin{align*}
-2(4x - 15) &= -8x + 30, \\
-3(5x^2 + 7x + 6) &= -15x^2 - 21x - 18, \\
-8x + 30 - 15x^2 - 21x - 18 &= -15x^2 - 29x + 12.
\end{align*}
\][/tex]
This does not match the target polynomial.
6. Expression 6: [tex]\( (-19x^2 - 4x - 7) + (4x^2 + 25x - 5) \)[/tex]
Expand and simplify:
[tex]\[
\begin{align*}
-19x^2 - 4x - 7 + 4x^2 + 25x - 5 &= (-19x^2 + 4x^2) + (-4x + 25x) + (-7 - 5) \\
&= -15x^2 + 21x - 12.
\end{align*}
\][/tex]
This does not match the target polynomial.
Based on our calculations, the expressions equivalent to the polynomial [tex]\( -15x^2 - 29x - 12 \)[/tex] are:
[tex]\[
\boxed{1, 2, 3}
\][/tex]
