Expand $(2z^2 + 5z - 6)(3z^3 - 2z + 1)$.
$$\begin{array}{crrrrrrr}
& & & 3z^3 & & -2z & + 1 & \\
\times & & & & 2z^2 & +5z & -6 \\
\cline{1-7}\rule{0pt}{0.17in}
& & & -18z^3 & & +12z & -6 & \\
& & +15z^4 & & -10z^2 & +5z & & \\
+ & 6z^5 & & -4z^3 & +2z^2 & & & \\
\cline{1-7}\rule{0pt}{0.17in}
& 6z^5 & +15z^4 & -22z^3 & - 8z^2 &+17z & -6 &
\end{array}$$ As such, the answer is $\boxed{6z^5+15z^4-22z^3-8z^2+17z-6}$.
6z^5+15z^4-22z^3-8z^2+17z-6
Level 5
Algebra
Problem: Expand $(2z^2 + 5z - 6)(3z^3 - 2z + 1)$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To expand the expression, apply the distributive property, also known as the FOIL method for polynomials. Start by distributing each term in the first polynomial, \(2z^2 + 5z - 6\), to each term in the second polynomial, \(3z^3 - 2z + 1\).

1. Distribute \(2z^2\):
   - \(2z^2 \cdot 3z^3 = 6z^5\)
   - \(2z^2 \cdot (-2z) = -4z^3\)
   - \(2z^2 \cdot 1 = 2z^2\)

2. Distribute \(5z\):
   - \(5z \cdot 3z^3 = 15z^4\)
   - \(5z \cdot (-2z) = -10z^2\)
   - \(5z \cdot 1 = 5z\)

3. Distribute \(-6\):
   - \(-6 \cdot 3z^3 = -18z^3\)
   - \(-6 \cdot (-2z) = 12z\)
   - \(-6 \cdot 1 = -6\)

Now, combine all the results:
- From \(2z^2\): \(6z^5\), \(15z^4\), \(-4z^3\), \(-10z^2\), \(5z\), \(-18z^3\), \(12z\), and \(-6\).

Next, group like terms:
- The \(z^5\) term: \(6z^5\)
- The \(z^4\) term: \(15z^4\)
- The \(z^3\) terms: \(-4z^3 - 18z^3 = -22z^3\)
- The \(z^2\) terms: \(2z^2 - 10z^2 = -8z^2\)
- The \(z\) terms: \(5z + 12z = 17z\)
- The constant term: \(-6\)

Finally, combine all the grouped terms to arrive at the final expanded expression:
The final answer is \(6z^5 + 15z^4 - 22z^3 - 8z^2 + 17z - 6\).<|END_OF_LATENT|> The final answer is $6z^5+15z^4-22z^3-8z^2+17z-6$.