https://file.aiquickdraw.com/mathbot/file/17229431191722xr5t2xz.jpg

The equation in the image is:

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

This is a quadratic equation in two variables, \( x \) and \( y \). To analyze this, we can try to rewrite it in a more standard form.

First, let's complete the square for both \( x \) and \( y \):

1. **For \( x \):**
   \[ x^2 + 2x \]
   Completing the square involves taking half the coefficient of \( x \), squaring it, and adding/subtracting it inside the equation:
   \[ x^2 + 2x = (x + 1)^2 - 1 \]

2. **For \( y \):**
   \[ y^2 + 3y \]
   Similarly, take half the coefficient of \( y \), square it, and add/subtract it:
   \[ y^2 + 3y = \left(y + \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 = \left(y + \frac{3}{2}\right)^2 - \frac{9}{4} \]

Now, substitute these back into the original equation:
\[ (x + 1)^2 - 1 + \left(y + \frac{3}{2}\right)^2 - \frac{9}{4} = 12 \]

Simplify this equation:
\[ (x + 1)^2 + \left(y + \frac{3}{2}\right)^2 - 1 - \frac{9}{4} = 12 \]
\[ (x + 1)^2 + \left(y + \frac{3}{2}\right)^2 - \frac{13}{4} = 12 \]
\[ (x + 1)^2 + \left(y + \frac{3}{2}\right)^2 = 12 + \frac{13}{4} \]
\[ (x + 1)^2 + \left(y + \frac{3}{2}\right)^2 = 12 + 3.25 \]
\[ (x + 1)^2 + \

Explore the detailed analysis of the equation x^2 + y^2 + 2x + 3y = 12 through completing the square. This problem covers algebraic concepts such as quadratic equations and completing the square method, suitable for students in Grades 9-12.

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