To find the best-fit curve of the form $p(x) = ax + bx^2$ for the data points provided, we can use the least squares method to determine the values of $a$ and $b$.

The data points given are: $(0, 0), (1, 5), (2, 1)$

Step 1: Set Up the Equations Using Least Squares

We need to find values of $a$ and $b$ that minimize the sum of the squared residuals: $\sum (y_i - (ax_i + bx_i^2))^2$

This can be expanded to a system of equations based on each data point. Specifically, we substitute each $(x_i, y_i)$ into $y = ax + bx^2$ and create a system of equations.

For our three points:

1. When $x = 0$ and $y = 0$: $0 = a \cdot 0 + b \cdot 0^2 \Rightarrow 0 = 0$

2. When $x = 1$ and $y = 5$: $5 = a \cdot 1 + b \cdot 1^2 \Rightarrow 5 = a + b$

3. When $x = 2$ and $y = 1$: $1 = a \cdot 2 + b \cdot 2^2 \Rightarrow 1 = 2a + 4b$

Thus, we get the following system of linear equations: