Use synthetic division to divide. (3x^2-7)/(x-2) (3x^2-7)/(x-2)= square

To use synthetic division to divide $\frac{3x^2 - 7}{x - 2}$, we first need to set up the synthetic division. The divisor is $x - we use \(2$ as the number for synthetic division.

Here are the steps:

1. Write down the coefficients of the dividend $3x^2 - 7$. Since there is no $x$ term, we include a coefficient of $0$:
$$3 \quad 0 \quad -7$$

2. Write $2$ to the left and draw a line:
$$\begin{array}{r|rrr} 2 & 3 & 0 & -7 \\ & & & \\ \hline & 3 & & \\ \end{array}$$

3. Bring down the first coefficient (3):
\[
\begin{array}{r|rr 2 & 3 & 0 & -7 \\
& & & \\
\hline
& 3 & & \\
\end{array}
\]

4. Multiply $2$ by the value just written below the line (3), and write the result under the next coefficient:
$$\begin{array}{r|rrr} 2 & 3 & 0 & -7 \\ & & 6 & \\ \hline & 3 & 6 & \\ \end{array}$$

5. Add the column of numbers:
$$\begin{array}{r|rrr} 2 & 3 & 0 & -7 \\ &6 & 12 \\ \hline & 3 & 6 & 5 \\ \end{array}$$

The bottom row gives us the coefficients of the quotient polynomial and the remainder. The quotient is $3x + 6$ and the remainder is $5$.

So, the result of the division is:
$$\frac{3x^2 - 7}{x - 2} = 3x + 6 + \frac{5}{x - 2}$$

Thus, the final answer is:
$$\boxed{3x + 6 + \frac{5}{x - 2}}$$