Identify each equation without completing the square.
$9x^2+4y^2-36x+8y+31=0$

Verified step by step guidance

1. Recognize the given equation: \(9x^2 + 4y^2 - 36x + 8y + 31 = 0\). This is a quadratic equation in two variables, which suggests it might represent a conic section.
2. Identify the coefficients of \(x^2\) and \(y^2\). Here, the coefficients are 9 and 4, respectively. Since both coefficients are positive and different, this indicates the equation represents an ellipse.
3. To confirm it's an ellipse, check if the equation can be rewritten in the standard form of an ellipse: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). This involves completing the square for both \(x\) and \(y\) terms.
4. For the \(x\) terms, factor out the coefficient of \(x^2\), which is 9, from the terms \(9x^2 - 36x\). This gives \(9(x^2 - 4x)\).
5. For the \(y\) terms, factor out the coefficient of \(y^2\), which is 4, from the terms \(4y^2 + 8y\). This gives \(4(y^2 + 2y)\).