To find the quadratic equation that passes through the given points $(-7, 134)$, $(-3, 10)$, and $(5, 50)$, we need to fit these points to a quadratic function of the form:

$y = ax^2 + bx + c$

Step 1: Set up the system of equations

For each point, substitute the $x$ and $y$ values into the general form $y = ax^2 + bx + c$:

1. For $(-7, 134)$: $134 = a(-7)^2 + b(-7) + c \quad \Rightarrow \quad 134 = 49a - 7b + c$

2. For $(-3, 10)$: $10 = a(-3)^2 + b(-3) + c \quad \Rightarrow \quad 10 = 9a - 3b + c$

3. For $(5, 50)$: $50 = a(5)^2 + b(5) + c \quad \Rightarrow \quad 50 = 25a + 5b + c$

Step 2: Solve the system of equations

We now have the system: