To find the quadratic regression curve for the given points $(-2, 6)$, $(-3, 5)$, $(-4, 3)$, and $(-5, 4)$, we need to fit a quadratic function of the form:

$y = ax^2 + bx + c$

We will use the method of least squares to determine the coefficients $a$, $b$, and $c$. Given the points $(-2, 6)$, $(-3, 5)$, $(-4, 3)$, and $(-5, 4)$, this process involves solving a system of equations that minimizes the sum of the squared differences between the observed and predicted values.

Let's calculate this quadratic regression now.

Steps:

1. Set up the system of equations based on the quadratic model.
2. Solve for the coefficients $a$, $b$, and $c$.
3. Round all coefficients to four decimal places.

Let me compute that for you.The quadratic regression curve that fits the given points is:

$y = 0.5x^2 + 4.3x + 12.8$

All coefficients have been rounded to four decimal places as required.

Would you like further details on the process or have any other questions? Here are five related questions to deepen understanding:

1. How does the least squares method work for quadratic regression?
2. How would you interpret each coefficient of the quadratic equation?
3. Can you calculate the predicted $y$-values for these specific $x$-values using this equation?
4. What is the difference between quadratic regression and linear regression?
5. How would you compute the residuals (errors) for each point?

Tip: For any regression analysis, the fit improves as the sum of squared residuals decreases.