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The image asks for the standard form of the equation of the ellipse given by the expression:

\[
9x^2 + 4y^2 + 54x - 16y + 61 = 0
\]

### Solution:

To find the standard form of an ellipse, we first need to complete the square for both the \(x\) and \(y\) terms.

1. **Group the \(x\) and \(y\) terms:**

\[
9x^2 + 54x + 4y^2 - 16y = -61
\]

2. **Factor out the coefficients of \(x^2\) and \(y^2\):**

\[
9(x^2 + 6x) + 4(y^2 - 4y) = -61
\]

3. **Complete the square:**

   For \(x\)-terms: \(x^2 + 6x\)
   - Take half of 6, square it: \((6/2)^2 = 9\)
   - Add and subtract 9 inside the parentheses:

\[
9(x^2 + 6x + 9 - 9) = 9((x + 3)^2 - 9)
\]

For \(y\)-terms: \(y^2 - 4y\)
   - Take half of -4, square it: \((-4/2)^2 = 4\)
   - Add and subtract 4 inside the parentheses:

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

Now, substitute these back into the equation:

\[
9((x + 3)^2 - 9) + 4((y - 2)^2 - 4) = -61
\]

Expand and simplify:

\[
9(x + 3)^2 - 81 + 4(y - 2)^2 - 16 = -61
\]

Combine constants:

\[
9(x + 3)^2 + 4(y - 2)^2 = 36
\]

4. **Divide through by 36 to achieve the standard form:**

\[
\frac{9(x + 3)^2}{36} + \frac{4(y - 2)^2}{36} = 1
\]

Simplify the fractions:

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

### Final Answer:

The standard form of the ellipse is:

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

### Follow-up Questions:

1. What is the center of the ellipse from the standard form equation?
2. What are the lengths of the semi-major and semi-minor axes of the ellipse?
3. How do you determine the orientation of the ellipse based on the standard form?
4. How would you find the foci of this ellipse?
5. What is the eccentricity of the ellipse, and how is it calculated?

**Tip:** Remember, completing the square is a key technique for transforming quadratic equations into a more usable form!

Determine the standard form of the equation of the ellipse given by the expression: 9x^2 + 4y^2 + 54x - 16y + 61 = 0

Learn how to convert the equation 9x^2 + 4y^2 + 54x - 16y + 61 = 0 into the standard form of an ellipse. This problem covers key algebraic concepts, including completing the square and the equation of an ellipse. Suitable for students in Grades 10-12.

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