The quadrilateral formed by the points $( -1, -2), (1, 0), (-1, 2)$, and $( -3, 0)$ is a:

Let $A (-1,-2), B (1,0) , C (-1,2)$ and $D (3,0)$.

Distance between two points $\left( { x }_{ 1 },{ y }_{ 1 } \right)$ and $\left( { x }_{ 2 },{ y }_{ 2 } \right)$ can be calculated using the formula $\sqrt { \left( { x }_{ 2 }-{ x }_{ 1 } \right) ^{ 2 }+\left( { y }_{ 2 }-{ y}_{ 1 } \right) ^{ 2 } }$

Distance between the points A$(-1,-2)$ and B $(1,0) = \sqrt { \left( 1+1\right) ^{ 2 }+\left( 0 +2 \right) ^{ 2 } } = \sqrt { 4 + 4 } = \sqrt { 8 }$

Distance between the points B$(1,0)$ and C $(-1,2) = \sqrt { \left( -1-1 \right) ^{ 2 }+\left( 2-0\right) ^{ 2 } } = \sqrt { 4 + 4 } = \sqrt { 8 }$

Distance between the points C$(-1,2)$ and D $(-3,0) = \sqrt { \left( -3 +1\right) ^{ 2 }+\left( 0-2\right) ^{ 2 } } = \sqrt {4 + 4 } = \sqrt { 8 }$

Distance between the points A$(-1,-2)$ and D $(-3,0) = \sqrt { \left( -3+1\right) ^{ 2 }+\left( 0 + 2\right) ^{ 2 } } = \sqrt { 4 + 4 } = \sqrt { 8 }$

Since, length of the sides between all vertices are equal, they are the vertices of a square.