Solve the System: x²+4=-6y and y²+9=-4x Equations

To determine the correct relationship between $x$ and $y$, let's transform each equation:

Step 1: Transform the First Equation

The first equation is $x^2 + 4 = -6y$. Rearranging gives us:

$x^2 = -6y - 4$

Now, aim to complete the square for expressions involving $x$ and $y$.

Step 2: Transform the Second Equation

The second equation is $y^2 + 9 = -4x$. Rearranging gives us:

$y^2 = -4x - 9$

Step 3: Complete the Square

Let's complete the square for the terms $x^2$ and $y^2$.

For $x^2 = -6y - 4$:

$x^2 + 4x + 4 = -6y$

Thus, it becomes:

$(x + 2)^2 = -6y$

For $y^2 = -4x - 9$:

$y^2 + 6y + 9 = -4x$

Thus, it becomes:

$(y + 3)^2 = -4x$

Step 4: Combine and Analyze

Substitute back to express a sum of squares:

Adding these completes the square:

$(x + 2)^2 + (y + 3)^2 = 0$

This result shows that both squares, squared terms are zero-sum, revealing the conditions under which equations balance.

Thus, the correct choice according to the transformations conducted is:

$(x+2)^2 + (y+3)^2 = 0$

Therefore, the solution to the problem is $(x+2)^2 + (y+3)^2 = 0$.