The equation you provided is:

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

To make this equation more recognizable, let's complete the square for both \(x\) and \(y\). This will help us rewrite the equation in a standard form.

### Step 1: Completing the square for \(x\)
The \(x\)-terms are \(x^2 + 4x\).

Take half of the coefficient of \(x\), which is 4, divide by 2 to get 2, then square it to get \(2^2 = 4\).

Add and subtract 4 inside the equation:

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

### Step 2: Completing the square for \(y\)
The \(y\)-terms are \(y^2 - 2y\).

Take half of the coefficient of \(y\), which is -2, divide by 2 to get -1, then square it to get \((-1)^2 = 1\).

Add and subtract 1 inside the equation:

\[
y^2 - 2y = (y - 1)^2 - 1
\]

### Step 3: Substituting back into the original equation
Now substitute the completed squares for both \(x\) and \(y\) into the original equation:

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

Simplify the equation:

\[
(x + 2)^2 + (y - 1)^2 - 5 = 4
\]

Add 5 to both sides:

\[
(x + 2)^2 + (y - 1)^2 = 9
\]

### Final Equation
The final equation is:

\[
(x + 2)^2 + (y - 1)^2 = 9
\]

This is the equation of a circle with:
- Center: \((-2, 1)\)
- Radius: \(3\) (since \(\sqrt{9} = 3\))

### Conclusion
The given equation represents a circle centered at \((-2, 1)\) with a radius of 3.

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Would you like any further details or have any questions about this solution?

Here are 5 related questions to explore further:
1. How do you complete the square for equations with larger coefficients?
2. What is the general equation of a circle, and how do you derive it?
3. How does completing the square help in graphing conic sections?
4. What are the applications of completing the square in solving quadratic equations?
5. How do you find the radius and center of a circle from its equation?

**Tip:** Completing the square is not just useful for finding the standard form of circles but also for solving quadratic equations and finding vertex forms of parabolas!

Learn how to rewrite the equation x^2 + 4x + y^2 - 2y = 4 in the standard form of a circle by completing the square. This problem covers essential algebra concepts such as completing the square and circle equations. Suitable for students in Grades 9-11.

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