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Final answer: $\boxed{2x^4 - 14x^2 + 23}$
Question: A comet follows a hyperbolic trajectory described by $ 4x^2 - 9y^2 - 32x + 54y + 115 = 0 $. Determine the center of the hyperbola.
Solution: Rewrite the equation by completing the square. Group $ x $ and $ y $ terms:
4(x^2 - 8x) - 9(y^2 - 6y) = -115.
Complete the square for $ x $: $ x^2 - 8x = (x - 4)^2 - 16 $.
For $ y $: $ y^2 - 6y = (y - 3)^2 - 9 $.
4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115 \implies 4(x - 4)^2 - 64 - 9(y - 3)^2 + 81 = -115.
4(x - 4)^2 - 9(y - 3)^2 + 17 = -115 \implies 4(x - 4)^2 - 9(y - 3)^2 = -132.
Divide by $ -132 $ to standardize:
\frac{(y - 3)^2}{\frac{132}{9}} - \frac{(x - 4)^2}{\frac{132}{4}} = 1.