The given equation is:

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

This is the equation of a circle, but it's not in standard form yet. To express it in standard form, we need to complete the square for both the \(x\) and \(y\) terms.

### Step 1: Complete the square for \(x\)
The \(x\)-terms are \(x^2 + 4x\). To complete the square, take half of the coefficient of \(x\) (which is 4), divide it by 2, and square it:

\[
\left(\frac{4}{2}\right)^2 = 4
\]

So, we add and subtract 4 to complete the square:

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

### Step 2: Complete the square for \(y\)
The \(y\)-terms are \(y^2 - 2y\). Similarly, take half of the coefficient of \(y\) (which is -2), divide it by 2, and square it:

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

So, we add and subtract 1 to complete the square:

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

### Step 3: Substitute back into the original equation
Substituting the completed squares back into the equation:

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

Simplify the constants:

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

Add 5 to both sides:

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

### Final Equation
The equation of the circle in standard form is:

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

This represents a circle with center \((-2, 1)\) and radius 3 (since \( \sqrt{9} = 3\)).

Would you like further details or have any questions?

Here are 5 related questions you can try:
1. What is the equation of a circle with center \((3, -4)\) and radius 5?
2. How do you convert \(x^2 + y^2 + 6x - 8y + 9 = 0\) to standard form?
3. Can you find the equation of a circle given three points on the circumference?
4. How do you find the distance between the center of a circle and a point on the circumference?
5. What is the general formula for the equation of a circle?

**Tip:** Completing the square is a useful technique for transforming quadratic equations into more interpretable forms.

Learn how to transform the equation x^2 + 4x + y^2 - 2y = 4 into the standard form of a circle by completing the square. This step-by-step guide explains key algebraic and geometric concepts. Suitable for students in Grades 9-11.

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