Let's solve and analyze the given equation:

\[ x^2 + y^2 + 2x + 3y = 12 \]

To rewrite this in a standard form, we need to complete the square for both \( x \) and \( y \) terms.

### Step-by-Step Solution

#### Completing the Square

1. **For \( x \)-terms:**
   - Original terms: \( x^2 + 2x \)
   - Completing the square: 
     \[
     x^2 + 2x + 1 - 1 = (x + 1)^2 - 1
     \]

2. **For \( y \)-terms:**
   - Original terms: \( y^2 + 3y \)
   - Completing the square: 
     \[
     y^2 + 3y + \left(\frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 = \left(y + \frac{3}{2}\right)^2 - \frac{9}{4}
     \]

#### Substitute back into the equation:

\[
(x + 1)^2 - 1 + \left(y + \frac{3}{2}\right)^2 - \frac{9}{4} = 12
\]

Simplify the constants on the right side:

\[
(x + 1)^2 + \left(y + \frac{3}{2}\right)^2 - 1 - \frac{9}{4} = 12
\]

Combine the constants:

\[
(x + 1)^2 + \left(y + \frac{3}{2}\right)^2 -

Learn how to solve the equation x^2 + y^2 + 2x + 3y = 12 using the method of completing the square. This problem involves key algebraic concepts, including quadratic equations and completing the square technique. Suitable for students in Grades 10-12.

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