The problem in the uploaded image appears to involve constructing a polynomial based on specific conditions. To help you better, I'll transcribe the problem and explain the solution step by step.

The text of the problem seems to read:

• Find a polynomial $P(x, y)$ such that $(-1, -1)$ and $(1, 0)$ are its roots, with the additional condition that the coefficient of $x^2$ is 1.

If this is correct, let's break it down:

Step 1: Form the polynomial

If $(-1, -1)$ and $(1, 0)$ are roots, then we know the polynomial must include factors corresponding to these roots:

1. $x + y + 1 = 0$ for $(-1, -1)$,
2. $x - y = 0$ for $(1, 0)$.

So, the polynomial can be expressed as the product of these two factors: $P(x, y) = (x + y + 1)(x - y).$

Step 2: Expand the polynomial

Expand $P(x, y)$ by multiplying the two factors: $P(x, y) = (x + y + 1)(x - y).$ $P(x, y) = x^2 - y^2 + x - y.$

Step 3: Ensure the coefficient of $x^2$ is 1

The coefficient of $x^2$ is already 1, so the polynomial remains unchanged: $P(x, y) = x^2 - y^2 + x - y.$

Final Answer:

$P(x, y) = x^2 - y^2 + x - y.$

Would you like additional clarification or steps explained in more detail? Here are 5 possible follow-up questions:

1. How can we verify that $(-1, -1)$ and $(1, 0)$ are roots of this polynomial?
2. Why does the condition about the coefficient of $x^2$ matter?
3. Can this method be generalized for other types of roots?
4. How would the solution change if additional roots were specified?
5. What does the polynomial represent geometrically in terms of $x$ and $y$?

Tip: Always expand and simplify polynomials carefully to avoid missing terms during multiplication!