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------------------------------------------------ What is the remainder in the synthetic division problem below?

[tex]\[

-2 \, \bigg| \begin{array}{llll}

2 & -3 & 1 \\

\end{array}

\][/tex]

Sure! Let's go through the synthetic division step-by-step to find the remainder.

Problem Statement:
We need to find the remainder when dividing the polynomial [tex]$2x^2 - 3x + 1$[/tex] by [tex]$x + 2$[/tex].

### Steps for Synthetic Division:

Step 1: Set up the problem
- Start with the coefficients of the polynomial: [tex]$2, -3, 1$[/tex].
- Use the opposite sign of the number added to [tex]$x$[/tex] in the divisor. Since the divisor is [tex]$x + 2$[/tex], we use [tex]$-2$[/tex].

Step 2: Bring down the first coefficient
- Bring down the first coefficient, which is [tex]$2$[/tex]. This is our starting remainder.

Step 3: Multiply and Add
- Multiply the current remainder by the divisor ([tex]$-2$[/tex]) and add this result to the next coefficient.
1. [tex]$2 \times (-2) = -4$[/tex]
2. Add this result to the next coefficient, [tex]$-3$[/tex]: [tex]$-3 + (-4) = -7$[/tex].

- Now, repeat this multiplication and addition with the new remainder:
1. [tex]$-7 \times (-2) = 14$[/tex]
2. Add this to the next coefficient, [tex]$1$[/tex]: [tex]$1 + 14 = 15$[/tex].

Step 4: Determine the remainder
- The last value obtained is the remainder of the division.

Thus, the remainder of the synthetic division is [tex]$15$[/tex].