Find the distance between the points `(-5, 7) and (-1, 3)`.

Answer

To find the distance between the points \((-5, 7)\) and \((-1, 3)\), we can use the distance formula. The distance \(d\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates of the points:**
- Let \(A = (-5, 7)\) where \(x_1 = -5\) and \(y_1 = 7\).
- Let \(B = (-1, 3)\) where \(x_2 = -1\) and \(y_2 = 3\).

2. **Substitute the values into the distance formula:**
\[ d = \sqrt{((-1) - (-5))^2 + (3 - 7)^2} \]

3. **Calculate \(x_2 - x_1\):**
\[ x_2 - x_1 = -1 + 5 = 4 \]

4. **Calculate \(y_2 - y_1\):**
\[ y_2 - y_1 = 3 - 7 = -4 \]

5. **Substitute these values back into the formula:**
\[ d = \sqrt{(4)^2 + (-4)^2} \]

6. **Calculate the squares:**
\[ d = \sqrt{16 + 16} \]

7. **Add the results:**
\[ d = \sqrt{32} \]

8. **Simplify \(\sqrt{32}\):**
\[ d = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

### Final Answer:
The distance between the points \((-5, 7)\) and \((-1, 3)\) is \(4\sqrt{2}\).