To find the quadratic regression curve through the given points $(0, 0), (-3, 3), (2, 2), (-5, 5)$, we use the quadratic model:

$y(x) = ax^2 + bx + c$

The general method for finding the coefficients $a$, $b$, and $c$ is to perform a least-squares quadratic regression using the points. The steps involve:

1. Setting up the system of equations based on the points.
2. Solving the normal equations derived from minimizing the sum of squared residuals.

We can use a technological tool (such as a calculator or software like Excel, Python, or a graphing calculator) to find the coefficients of the quadratic equation.

Let's compute the quadratic regression using technology.The quadratic regression curve that fits the points $(0, 0)$, $(-3, 3)$, $(2, 2)$, and $(-5, 5)$ is:

$y(x) = 0.2x^2 + 0.0828x + 0.7241$

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. How do you solve a system of linear equations to find regression coefficients?
2. What is the method of least squares, and why is it used in regression?
3. Can quadratic regression be used to predict data for more points?
4. How do you interpret the coefficients $a$, $b$, and $c$ in a quadratic equation?
5. How does quadratic regression differ from linear regression?

Tip: In regression problems, outliers (points that don't follow the general trend) can significantly affect the results.