Given the information about the quadratic inequality, we know the following:
- The vertex of the quadratic is at \((-1, -1)\).
- The quadratic opens upward.
- It passes through the point \((0, 1)\).
- The inequality represents the region below the curve.
Let's express the quadratic in vertex form, which is given by:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex. Substituting the vertex \((-1, -1)\):
\[ f(x) = a(x + 1)^2 - 1 \]
Next, we need to determine the value of \(a\). We know that the quadratic passes through the point \((0, 1)\):
\[ f(0) = a(0 + 1)^2 - 1 = 1 \]
Simplifying this equation:
\[ a(1)^2 - 1 = 1 \]
\[ a - 1 = 1 \]
\[ a = 2 \]
Now we can write the quadratic function:
\[ f(x) = 2(x + 1)^2 - 1 \]
To express it in standard form:
\[ f(x) = 2(x^2 + 2x + 1) - 1 = 2x^2 + 4x + 2 - 1 = 2x^2 + 4x + 1 \]
Since the shading is below the curve (the region where the quadratic is less than or equal to zero), the quadratic inequality is:
\[ 2x^2 + 4x + 1 \leq 0 \]
Thus, the quadratic inequality that represents the description is:
\[ \boxed{2x^2 + 4x + 1 \leq 0} \]