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The problem asks to express the quadratic function \( f(x) = x^2 + 2x - 4 \) in the form \( f(x) = (x + m)^2 - n \), where \( m \) and \( n \) are constants. Let's solve for \( m \) and \( n \).

### Step-by-step solution:

We start with the given quadratic function:

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

#### Step 1: Complete the square.
To express this quadratic in the form \( (x + m)^2 - n \), we need to complete the square.

The expression is \( x^2 + 2x - 4 \).

- Take the coefficient of \( x \), which is 2, and divide it by 2: \( \frac{2}{2} = 1 \).
- Square the result: \( 1^2 = 1 \).

Now, we add and subtract 1 inside the quadratic expression:

\[
f(x) = (x^2 + 2x + 1) - 1 - 4
\]

This simplifies to:

\[
f(x) = (x + 1)^2 - 5
\]

#### Step 2: Compare the form.
Now, the expression is in the form \( f(x) = (x + m)^2 - n \).

By comparing \( (x + 1)^2 - 5 \) with \( (x + m)^2 - n \), we get:

- \( m = 1 \)
- \( n = 5 \)

### Final Answer:
- \( m = 1 \)
- \( n = 5 \)

Would you like any further clarification or details?

Here are some related questions you can explore:
1. How do you complete the square for a quadratic function with a coefficient other than 1?
2. What is the geometric interpretation of completing the square for a quadratic function?
3. How do you factor a quadratic equation that cannot be completed into a perfect square?
4. How does completing the square help in solving quadratic equations?
5. How can the vertex form of a quadratic function be derived from standard form?

**Tip:** Completing the square is useful for converting quadratic equations into vertex form, which can help easily identify the maximum or minimum points of the function.

The quadratic function f(x) = x^2 + 2x - 4 can be expressed in the form f(x) = (x + m)^2 - n, where m and n are constants. Find the values of m and n.

Learn how to express the quadratic function f(x) = x^2 + 2x - 4 in the form (x + m)^2 - n by completing the square. This step-by-step solution demonstrates how to find the values of m and n, making it ideal for Grades 9-10 students.

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