To find the quadratic regression curve through the given points $(0, 1), (-3, 5), (-1, 3), (-5, 4)$, we can use the method of least squares to fit a quadratic model of the form $y = ax^2 + bx + c$. Let's calculate the coefficients $a$, $b$, and $c$ using Python.

Here are the steps we will follow:

1. Set up the matrix equations for least squares fitting.
2. Solve for the coefficients $a$, $b$, and $c$.

I'll perform these calculations now.The quadratic regression curve through the given points $(0, 1), (-3, 5), (-1, 3), (-5, 4)$ is:

$y(x) = -0.3631x^2 - 2.4209x + 0.9799$

Would you like more details or have any questions about this solution?

Here are some related questions you might consider:

1. How do you derive the least squares method for quadratic regression?
2. Can you explain the process of setting up the design matrix?
3. How does the least squares method minimize the error?
4. What are some practical applications of quadratic regression?
5. How can we visualize the fitted quadratic curve along with the given points?

Tip: When fitting regression models, always check the residuals to ensure the model's assumptions are met and the fit is appropriate.