College

Choose the correct simplification of \((6x - 5)(2x^2 - 3x - 6)\).

A. \(12x^3 + 28x^2 + 21x + 30\)

B. \(12x^3 - 28x^2 - 21x + 30\)

C. \(12x^3 + 28x^2 - 21x + 30\)

D. \(12x^3 - 28x^2 - 21x - 30\)

Answer :

We want to simplify
$$
(6x - 5)\left(2x^2 - 3x - 6\right).
$$

**Step 1: Distribute each term from the first polynomial to each term in the second polynomial.**

Multiply the term $6x$ by each term in the second polynomial:
\[
\begin{aligned}
6x \cdot 2x^2 &= 12x^3,\\[1mm]
6x \cdot (-3x) &= -18x^2,\\[1mm]
6x \cdot (-6) &= -36x.
\end{aligned}
\]

Next, multiply the term $-5$ by each term in the second polynomial:
\[
\begin{aligned}
-5 \cdot 2x^2 &= -10x^2,\\[1mm]
-5 \cdot (-3x) &= 15x,\\[1mm]
-5 \cdot (-6) &= 30.
\end{aligned}
\]

**Step 2: Combine like terms.**

Now, add the results from both distributions:

- The $x^3$ term:
$$
12x^3 \quad (\text{only one term}).
$$

- The $x^2$ terms:
$$
-18x^2 + (-10x^2) = -28x^2.
$$

- The $x$ terms:
$$
-36x + 15x = -21x.
$$

- The constant term:
$$
30.
$$

**Step 3: Write the final simplified expression.**

The simplified expression is:
$$
12x^3 - 28x^2 - 21x + 30.
$$

**Conclusion:**

The correct answer is
$$
\boxed{12x^3 - 28x^2 - 21x + 30}.
$$