The problem asks to find the best approximation of the form $y(x) = a + b(x^2 - x)$ to the data set $(-1, 1), (0, -1), (1, 3), (2, -5)$. We need to find the values of $a$ and $b$ that minimize the error between the given data points and the predicted values from the function.

This is essentially a curve-fitting or regression problem, where the goal is to minimize the sum of squared residuals:

$\text{Residual} = \sum_{i} \left( y_i - \left( a + b(x_i^2 - x_i) \right) \right)^2$

Steps:

1. We have the form $y(x) = a + b(x^2 - x)$, and we need to substitute the given points to create a system of equations.
2. The points given are:
   • $(-1, 1)$
   • $(0, -1)$
   • $(1, 3)$
   • $(2, -5)$

For each point $(x_i, y_i)$, we can substitute into the equation $y(x) = a + b(x^2 - x)$, and this will give us four linear equations in terms of $a$ and $b$.

3. Solving the system of equations will give the values of $a$ and $b$.

Let me set up and solve the system for you.It seems that directly solving the system did not yield a result. This might be due to the complexity of handling the approximation in a single step.

Let's adjust our approach to find the best-fitting values for $a$ and $b$ using least squares regression, which will minimize the sum of squared differences between the given data points and the model's predictions.

Let me calculate that next.The best approximation for the equation $y(x) = a + b(x^2 - x)$ to fit the given data set is:

$a = 1, \quad b = -1.5$

This means the equation is:

$y(x) = 1 - 1.5(x^2 - x)$

The sum of squared residuals, which measures the error between the fitted curve and the data points, is 26.

Would you like further details on how the least squares method works, or any other clarifications?

Here are some related questions:

1. How does least squares regression work in curve fitting?
2. Can this method be extended to other forms of regression equations?
3. What is the significance of minimizing the residuals in regression?
4. How would the solution change if we added more data points?
5. What if we use a different form for the approximating function?

Tip: Least squares regression is often used for fitting both linear and non-linear models to data, and it works by minimizing the sum of squared differences between observed and predicted values.