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The given problem involves a quadratic curve and a straight line with the following equations:

1. Quadratic curve: \( y = 4 - x - x^2 \)
2. Straight line: \( y = 2 \)

### Part (a): Find the coordinates of points \( A \) and \( B \)

To find the coordinates of \( A \) and \( B \), we need to determine the points of intersection between the quadratic curve and the straight line. This requires solving the system of equations by setting them equal to each other:

\[
4 - x - x^2 = 2
\]

Simplify the equation:

\[
4 - x - x^2 = 2 \implies -x - x^2 = 2 - 4 \implies -x - x^2 = -2
\]

Multiply through by \(-1\) to make the quadratic equation easier to solve:

\[
x + x^2 = 2 \implies x^2 + x - 2 = 0
\]

Now, factor the quadratic equation:

\[
(x - 1)(x + 2) = 0
\]

So, the solutions for \(x\) are:

\[
x = 1 \quad \text{or} \quad x = -2
\]

These are the \(x\)-coordinates of points \(A\) and \(B\).

Next, we substitute these \(x\)-values into the equation of the line \(y = 2\) (since both points lie on the line) to find the corresponding \(y\)-coordinates. Since the line is \(y = 2\), both points have the same \(y\)-value:

\[
y = 2
\]

Thus, the coordinates of points \(A\) and \(B\) are:
- \(A(-2, 2)\)
- \(B(1, 2)\)

### Part (b): Determine the exact area of the shaded region

The area of the finite region between the curve \(y = 4 - x - x^2\) and the line \(y = 2\) from \(x = -2\) to \(x = 1\) can be calculated using definite integration. The area is given by:

\[
\text{Area} = \int_{-2}^{1} \left[ (4 - x - x^2) - 2 \right] \, dx
\]

Simplify the integrand:

\[
(4 - x - x^2) - 2 = 2 - x - x^2
\]

Now, integrate this expression:

\[
\text{Area} = \int_{-2}^{1} (2 - x - x^2) \, dx
\]

Let me calculate this integral for you.The exact area of the finite region bounded by the quadratic curve \( y = 4 - x - x^2 \) and the straight line \( y = 2 \) is \( \frac{9}{2} \) square units.

### Summary:
- The coordinates of points \( A \) and \( B \) are \( A(-2, 2) \) and \( B(1, 2) \).
- The area of the shaded region is \( \frac{9}{2} \) square units.

Do you need any additional explanations or details on any part?

Here are 5 related questions for further understanding:
1. How would the area change if the line were \( y = 3 \) instead of \( y = 2 \)?
2. How can you solve a quadratic equation using the quadratic formula?
3. How do you verify the points of intersection geometrically?
4. What is the importance of definite integrals in finding areas under curves?
5. Can you find the maximum value of the quadratic function \( y = 4 - x - x^2 \)?

**Tip:** When solving quadratic equations, always check if factoring is possible first; it's often the quickest method!

The figure shows a quadratic curve and a straight line with the respective equations y = 4 - x - x^2 and y = 2. The points A and B are the points of intersection between the quadratic curve and the straight line. a) Find the coordinates of A and B. b) Determine the exact area of the finite region bounded by the quadratic curve and the straight line.

Solve a problem involving the intersection of a quadratic curve (y = 4 - x - x^2) and a straight line (y = 2). Find the coordinates of points A and B, and calculate the exact area of the shaded region using definite integration.

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