$ 4(x+3)^2 - 36 - 9(y-1)^2 + 9 = -9 $.

["Title: Simplifying and Analyzing the Equation: $4(x+3)^2 - 36 - 9(y-1)^2 + 9 = -9$", "---", "Introduction\nUnderstanding algebraic equations is fundamental in mathematics, especially when working with conic sections and coordinate geometry. The equation\n$$\n4(x+3)^2 - 36 - 9(y-1)^2 + 9 = -9\n$$\nmight appear complex at first glance, but with systematic simplification, we can identify its geometric meaning and key features. This SEO-optimized article walks you through solving, simplifying, and interpreting this equation for better comprehension and practical application.", "---", "Step 1: Simplify the Equation", "Start by combining like terms on the left-hand side:\n$$\n4(x+3)^2 - 9(y-1)^2 - 27 = -9\n$$\nAdd 27 to both sides to isolate the squared terms:\n$$\n4(x+3)^2 - 9(y-1)^2 = 18\n$$", "---", "Step 2: Normalize the Equation", "Divide the entire equation by 18 to put it into standard form:\n$$\n\frac{4(x+3)^2}{18} - \frac{9(y-1)^2}{18} = 1\n$$\nSimplify the fractions:\n$$\n\frac{(x+3)^2}{4.5} - \frac{(y-1)^2}{2} = 1\n$$", "---", "Step 3: Recognize the Conic Section Pattern", "The simplified equation matches the standard form of a hyperbola opening horizontally:\n$$\n\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\n$$\nwhere\n- Center: $(h,k) = (-3, 1)$\n- $a^2 = 4.5 \Rightarrow a = \frac{3}{\sqrt{2}}$\n- $b^2 = 2 \Rightarrow b = \sqrt{2}$", "---", "Step 4: Interpret the Geometry and Significance", "This hyperbola:\n- Has a horizontal transverse axis\n- Center located at $(-3, 1)$\n- Vertices at $(-3 \pm \frac{3}{\sqrt{2}}, 1)$\n- Asymptotes defined by the lines $y - 1 = \pm\frac{b}{a}(x + 3) = \pm\frac{\sqrt{2}}{3/\sqrt{2}}(x+3) = \pm\frac{2}{3}(x+3)$", "Understanding this shape helps visualize how data sets may be modeled using hyperbolic relationships or when analyzing constraints in optimization problems.", "---", "Conclusion", "The equation $4(x+3)^2 - 36 - 9(y-1)^2 + 9 = -9$ simplifies to a standard hyperbola centered at $(-3, 1)$ with distinct transverse and conjugate axes. Recognizing this equation’s structure enables deeper analysis in coordinate geometry, algebraic modeling, and applied mathematics. Whether solving equations or graphing conics, mastering transformations and standard forms like this one is key to advancing your mathematical proficiency.", "---", "SEO Keywords:\n$4(x+3)^2 - 9(y-1)^2 = 18$, hyperbola center at (-3,1), standard form hyperbola, algebraic equation simplification, conic sections analysis, center and asymptotes of hyperbola, coordinate geometry, solving quadratic equations, hyperbola vertex calculation", "Meta Title: Simplifying and Analyzing the Equation $4(x+3)^2 - 9(y-1)^2 = 18$ — Hyperbola Geometry Guide\nMeta Description: Learn how to simplify and interpret the equation $4(x+3)^2 - 36 - 9(y-1)^2 + 9 = -9$. Discover its hyperbolic form, center, and key geometric features to enhance your algebra skills."]