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The given expression is:

\[
-x^2 + x + 2
\]

We need to rewrite this in the form:

\[
a(x + b)^2 + c
\]

### Step 1: Factor out the coefficient of \(x^2\)
First, factor out the negative sign from the \(x^2\) term:

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

### Step 2: Complete the square
Next, complete the square inside the parentheses. To complete the square for \(x^2 - x\), take half of the coefficient of \(x\) (which is \(-1\)), square it, and add and subtract it:

\[
x^2 - x = \left(x^2 - x + \frac{1}{4}\right) - \frac{1}{4}
\]

So the expression becomes:

\[
-(x^2 - x) + 2 = -\left(\left(x - \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 2
\]

### Step 3: Simplify
Now distribute the negative sign and simplify:

\[
-\left(\left(x - \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 2 = -\left(x - \frac{1}{2}\right)^2 + \frac{1}{4} + 2
\]

Finally, combine the constants:

\[
\frac{1}{4} + 2 = \frac{1}{4} + \frac{8}{4} = \frac{9}{4}
\]

Thus, the expression becomes:

\[
-(x - \frac{1}{2})^2 + \frac{9}{4}
\]

### Step 4: Identify \(a\), \(b\), and \(c\)
From the final expression, we have:

\[
a = -1, \quad b = \frac{1}{2}, \quad c = \frac{9}{4}
\]

Therefore, the expression in the desired form is:

\[
-(x - \frac{1}{2})^2 + \frac{9}{4}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you complete the square for a quadratic expression?
2. What is the importance of factoring when rewriting quadratic expressions?
3. Can the method used here be applied to any quadratic equation?
4. How do you handle cases where the quadratic term has a coefficient other than 1 or -1?
5. What does completing the square tell us about the vertex of a parabola?

### Tip:
When completing the square, always check that you correctly balance the equation by adding and subtracting the same value inside the square.

Write -x^2 + x + 2 in the form a(x + b)^2 + c, where a, b, and c are rational numbers.

Learn how to rewrite the quadratic expression -x^2 + x + 2 in the form a(x + b)^2 + c by completing the square. This step-by-step solution covers key algebraic concepts, including completing the square and working with quadratic equations. Suitable for students in Grades 9-11.

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