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 will use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

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

### Step 1: Identify the coordinates
Let the first point \(A\) be \((-5, 7)\) and the second point \(B\) be \((-1, 3)\). Here, we have:
- \(x_1 = -5\)
- \(y_1 = 7\)
- \(x_2 = -1\)
- \(y_2 = 3\)

### Step 2: Substitute the coordinates into the distance formula
Now, we substitute the values into the distance formula:

\[ d = \sqrt{((-1) - (-5))^2 + (3 - 7)^2} \]

### Step 3: Simplify the expressions inside the square root
Calculate \(x_2 - x_1\) and \(y_2 - y_1\):

\[ x_2 - x_1 = -1 + 5 = 4 \]
\[ y_2 - y_1 = 3 - 7 = -4 \]

### Step 4: Square the differences
Now, we square the results:

\[ d = \sqrt{(4)^2 + (-4)^2} \]

Calculating the squares:

\[ d = \sqrt{16 + 16} \]

### Step 5: Add the squares
Now, add the squared values:

\[ d = \sqrt{32} \]

### Step 6: Simplify the square root
We can simplify \(\sqrt{32}\):

\[ d = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

### Final Answer
Thus, the distance between the points \((-5, 7)\) and \((-1, 3)\) is:

\[ \boxed{4\sqrt{2}} \]