To find the distance between the points $(0, -2)$ and $(-4, -10)$, we use the distance formula:

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

Here, the coordinates of the two points are:

• $(x_1, y_1) = (0, -2)$
• $(x_2, y_2) = (-4, -10)$

Now, applying the formula:

$d = \sqrt{((-4) - 0)^2 + ((-10) - (-2))^2}$

Let's calculate that step by step:

1. $(-4) - 0 = -4$
2. $(-10) - (-2) = -10 + 2 = -8$

So, the formula becomes:

$d = \sqrt{(-4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80}$

Now, simplify $\sqrt{80}$:

$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$

Thus, the distance between the two points is:

$d = 4\sqrt{5} \text{ units}$

Would you like more details or further clarification on any step?

Here are 5 related questions for practice:

1. Find the distance between the points $(1, 3)$ and $(4, -1)$.
2. Calculate the distance between $(-5, -7)$ and $(3, 2)$.
3. Determine the length of the diagonal of a rectangle with vertices at $(0, 0)$ and $(5, 12)$.
4. How do you calculate the distance between two points in three-dimensional space?
5. What is the relationship between the distance formula and the Pythagorean theorem?

Tip: Always simplify the radical in distance calculations whenever possible for the most accurate expression.