To find the equation of a parabola that passes through the points $(-2, 0)$, $(3, 0)$, and $(0, 6)$, we can use the general form of a quadratic equation:

$y = ax^2 + bx + c$

Since the points $(-2, 0)$ and $(3, 0)$ are the roots of the parabola, we know the equation can be written in factored form as:

$y = a(x + 2)(x - 3)$

Next, we will use the point $(0, 6)$ to determine the value of $a$. Substituting $x = 0$ and $y = 6$ into the factored form:

$6 = a(0 + 2)(0 - 3)$

$6 = a(2)(-3)$

$6 = -6a$

Solving for $a$:

$a = -1$

Thus, the equation of the parabola is:

$y = -(x + 2)(x - 3)$

We can expand this to get the standard form:

$y = -(x^2 - x - 6)$

$y = -x^2 + x + 6$

Therefore, the equation of the parabola is:

$y = -x^2 + x + 6$

Would you like a detailed explanation of the expansion, or have any further questions?

Here are some related questions:

1. How do you convert between standard form and factored form of a quadratic equation?
2. How do you find the vertex of a parabola given its equation?
3. How do you calculate the axis of symmetry for a parabola?
4. What is the geometric interpretation of the coefficient $a$ in a parabola's equation?
5. How do you determine the direction of opening for a parabola?

Tip: The sign of the coefficient $a$ in a parabola’s equation tells you whether it opens upward (positive $a$) or downward (negative $a$).