Let's perform the synthetic division step-by-step to find the number that belongs in the place of the question mark. We are given the problem:
```
-1 | 1 3 -5 7
| -1 ?
| 1 2 ? ?
```
The given divisor is -1, and the coefficients of the polynomial are 1, 3, -5, and 7. Follow these steps to complete the synthetic division:
1. Step 1: Bring down the leading coefficient (1).
```
-1 | 1 3 -5 7
|
| 1
```
2. Step 2: Multiply the divisor (-1) by the leading coefficient (1) and add to the next coefficient (3).
[tex]\[ -1 \times 1 = -1 \][/tex]
[tex]\[ 3 + (-1) = 2 \][/tex]
```
-1 | 1 3 -5 7
| -1
| 1 2
```
3. Step 3: Multiply the divisor (-1) by the new coefficient (2) and add to the next coefficient (-5).
[tex]\[ -1 \times 2 = -2 \][/tex]
[tex]\[ -5 + (-2) = -7 \][/tex]
```
-1 | 1 3 -5 7
| -1 -2
| 1 2 -7
```
4. Step 4: Multiply the divisor (-1) by the new coefficient (-7) and add to the next coefficient (7).
[tex]\[ -1 \times -7 = 7 \][/tex]
[tex]\[ 7 + 7 = 14 \][/tex]
```
-1 | 1 3 -5 7
| -1 -2 7
| 1 2 -7 14
```
The number that belongs in place of the question mark is -7.
