To determine whether the given point $(-4, 1)$ lies inside, outside, or on the circle defined by the equation $x^2 + y^2 + 2x + 8y - 13 = 0$, we need to compare the point's distance from the circle's center with the circle's radius.

Step 1: Rewrite the Circle's Equation in Standard Form

The equation of the circle is given as: $x^2 + y^2 + 2x + 8y - 13 = 0$

We will complete the square for both $x$ and $y$ terms:

1. For the $x$ terms: $x^2 + 2x$

   • Add and subtract $\left(\frac{2}{2}\right)^2 = 1$, so it becomes $(x + 1)^2 - 1$.

2. For the $y$ terms: $y^2 + 8y$

   • Add and subtract $\left(\frac{8}{2}\right)^2 = 16$, so it becomes $(y + 4)^2 - 16$.

Substituting these back into the equation: $(x + 1)^2 - 1 + (y + 4)^2 - 16 - 13 = 0$

Simplify the equation: [ (x + 1)^2 + (y + 4)^2 - 30 = 0