The problem presents two inequalities and one equation. The first inequality involves vector magnitudes: $18 \le ||\vec{MA} + 2\vec{MB} - \vec{MC}|| \le \sqrt{2} ||3\vec{MA} + 2\vec{MB} - 5\vec{MC}||$. The equation is $y^2 + (x+1)^2 = 25$. We need to solve these. It is not entirely clear what we are supposed to do with the inequalities, as there's no specific question. We can work with the equation $y^2 + (x+1)^2 = 25$.

1. Problem Description

The problem presents two inequalities and one equation. The first inequality involves vector magnitudes: $18 \le ||\vec{MA} + 2\vec{MB} - \vec{MC}|| \le \sqrt{2} ||3\vec{MA} + 2\vec{MB} - 5\vec{MC}||$. The equation is $y^2 + (x+1)^2 = 25$. We need to solve these. It is not entirely clear what we are supposed to do with the inequalities, as there's no specific question. We can work with the equation $y^2 + (x+1)^2 = 25$.

2. Solution Steps

The equation $y^2 + (x+1)^2 = 25$ represents a circle.

Equation of a circle:

$(x - h)^2 + (y - k)^2 = r^2$

where $(h, k)$ is the center of the circle and $r$ is the radius.

Comparing $y^2 + (x+1)^2 = 25$ with the general equation of a circle, we can rewrite the given equation as

$(x - (-1))^2 + (y - 0)^2 = 5^2$.

Therefore, the center of the circle is $(-1, 0)$ and the radius is $5$.

3. Final Answer

The equation $y^2 + (x+1)^2 = 25$ represents a circle with center $(-1, 0)$ and radius $5$.