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To express the quadratic equation \(-x^2 - 4x + 9\) in the form \(a(x + b)^2 + c\), we need to complete the square.

### Step-by-step process:

1. **Factor out the coefficient of \(x^2\):**
   Since the coefficient of \(x^2\) is negative, factor \(-1\) from the first two terms:
   \[
   -x^2 - 4x + 9 = -\left(x^2 + 4x\right) + 9
   \]

2. **Complete the square inside the parentheses:**
   To complete the square for \(x^2 + 4x\), take half the coefficient of \(x\) (which is 4), square it, and add and subtract it inside the parentheses:
   \[
   x^2 + 4x = \left(x^2 + 4x + 4\right) - 4 = (x + 2)^2 - 4
   \]
   Now substitute this back into the expression:
   \[
   -\left((x + 2)^2 - 4\right) + 9
   \]

3. **Simplify the expression:**
   Distribute the \(-1\) and simplify:
   \[
   -\left((x + 2)^2 - 4\right) + 9 = -(x + 2)^2 + 4 + 9 = -(x + 2)^2 + 13
   \]

Thus, the expression in the form \(a(x + b)^2 + c\) is:
\[
-(x + 2)^2 + 13
\]
where \(a = -1\), \(b = 2\), and \(c = 13\).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you complete the square when the coefficient of \(x^2\) is not 1?
2. What are the benefits of expressing a quadratic in vertex form \(a(x + b)^2 + c\)?
3. Can every quadratic function be expressed in the form \(a(x + b)^2 + c\)?
4. How can this process be applied to solve quadratic equations?
5. How does completing the square help in graphing quadratic functions?

**Tip:** Completing the square is especially useful in determining the vertex of a quadratic function, which helps in graphing the parabola accurately.

Express -x^2 - 4x + 9 in the form a(x + b)^2 + c

Learn how to convert the quadratic expression -x^2 - 4x + 9 into the form a(x + b)^2 + c using the method of completing the square. This step-by-step guide is perfect for high school students and covers key algebraic concepts.

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