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The problem asks to find the best fit of the form \( p(x) = ax + bx^2 \) for the given data set.

### Data points provided:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & -3 \\
0 & 0 \\
1 & 5 \\
2 & 1 \\
\hline
\end{array}
\]

### Objective:
We need to determine the values of \( a \) and \( b \) such that the quadratic model \( p(x) = ax + bx^2 \) best fits the data points.

This can be achieved by setting up a system of linear equations based on the least squares method or using a direct approach by plugging the data into the model and solving for \( a \) and \( b \).

Let me calculate this for you. I will proceed by solving the system to find \( a \) and \( b \).

### Model: 
\( y = ax + bx^2 \)

For the four given points, we generate the following system of equations:
1. \( -3 = a(-1) + b(-1)^2 \Rightarrow -3 = -a + b \)
2. \( 0 = a(0) + b(0^2) \Rightarrow 0 = 0 \)
3. \( 5 = a(1) + b(1^2) \Rightarrow 5 = a + b \)
4. \( 1 = a(2) + b(2^2) \Rightarrow 1 = 2a + 4b \)

Now, we will solve this system for \( a \) and \( b \).

Let's first simplify the system:

1. \( -3 = -a + b \Rightarrow a - b = 3 \)  (Equation 1)
2. \( 5 = a + b \)  (Equation 2)
3. \( 1 = 2a + 4b \)  (Equation 3)

We will solve these equations now:

### Solution Process:
From Equation (2): 
\[ a + b = 5 \Rightarrow a = 5 - b \]

Substitute this into Equation (1):
\[
(5 - b) - b = 3 \\
5 - 2b = 3 \\
2 = 2b \\
b = 1
\]

Substitute \( b = 1 \) into Equation (2):
\[
a + 1 = 5 \\
a = 4
\]

Thus, the best-fit model is:
\[
p(x) = 4x + x^2
\]

Would you like more details on how this fits the data or any further clarifications? 

Here are 5 related questions to extend your understanding:
1. How does the least squares method work in finding the best-fit model?
2. What would happen if we used a different form, like \( p(x) = a + bx + cx^2 \)?
3. How can we determine the accuracy of this quadratic fit?
4. What tools could be used to automate the solving of such systems?
5. How do changes in data points affect the values of \( a \) and \( b \)?

**Tip**: To solve systems of linear equations, you can also use matrix methods like Gaussian elimination or Cramer's rule, especially for larger systems.

Find the best fit of the form p(x) = ax + bx^2 to the data (note, not a + bx or a + bx + cx^2). Data: x = [-1, 0, 1, 2], y = [-3, 0, 5, 1].

Learn how to find the best-fit quadratic function of the form p(x) = ax + bx^2 for the data set x = [-1, 0, 1, 2] and y = [-3, 0, 5, 1]. This involves solving a system of linear equations using algebraic methods.

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