Divide using synthetic division.

To divide \(2x^3 + 3x^2 - 17x - 30\) by \(x - 2\) using synthetic division, we will follow these steps:

1. Set up synthetic division with the root of \(x - 2\), which is \(2\).
2. Write down the coefficients of the polynomial: \(2, 3, -17, -30\).
3. Carry down the leading coefficient and perform the synthetic division process.

Step 1: Set up synthetic division

The coefficients are:

• 2 (for \(2x^3\))
• 3 (for \(3x^2\))
• -17 (for \(-17x\))
• -30 (for the constant term)

Step 2: Synthetic division calculation

      2 |  2   3   -17  -30
        |       4    14   -6
        ---------------------
          2   7    -3   -36

1. Bring down the \(2\).
2. Multiply \(2\) (the number we are dividing by) by \(2\) (which we just brought down): \(2 \times 2 = 4\), and add it to \(3\): \(3 + 4 = 7\).
3. Multiply \(2\) by \(7\): \(2 \times 7 = 14\), and add it to \(-17\): \(-17 + 14 = -3\).
4. Multiply \(2\) by \(-3\): \(2 \times -3 = -6\), and add it to \(-30\): \(-30 - 6 = -36\).

Step 3: Write the result

The synthetic division has provided us with a quotient of \(2x^2 + 7x - 3\) and a remainder of \(-36\). So, we can express the result as:

\[ 2x^2 + 7x - 3 - \frac{36}{x - 2} \]

Now we can compare this result with the choices given:

• A: \(2x^2 + 7x - 3 + \frac{36}{x - 2}\)
• B: \(2x^2 - x - 15\)
• C: \(2x^2 + 7x - 3 - \frac{36}{x - 2}\) (This matches our result!)
• D: \(2x^2 + 7x + 3 - \frac{24}{x - 2}\)

The correct choice is C:
\[ \boxed{C} \]
which represents the result of the synthetic division accurately.