The sum of all real $x$ such that $\frac{4x^2 + 15x + 17}{x^2 + 4x + 12} = \frac{5x^2 + 16x + 18}{2x^2 + 5x + 13}$ is:
Step-by-Step Solution
Key Concept: Cross-multiply the rational equation and use Vieta's formulas on the resulting polynomial to find the sum of roots without finding individual roots.
Cross-multiply to get $(4x^2 + 15x + 17)(2x^2 + 5x + 13) = (5x^2 + 16x + 18)(x^2 + 4x + 12)$. Expanding the left side: $8x^4 + 20x^3 + 52x^2 + 30x^3 + 75x^2 + 195x + 34x^2 + 85x + 221 = 8x^4 + 50x^3 + 161x^2 + 280x + 221$. Expanding the right side: $5x^4 + 20x^3 + 60x^2 + 16x^3 + 64x^2 + 192x + 18x^2 + 72x + 216 = 5x^4 + 36x^3 + 142x^2 + 264x + 216$. Subtracting gives $3x^4 + 14x^3 + 19x^2 + 16x + 5 = 0$. Testing rational roots or factoring yields roots whose sum by Vieta's formulas is $-\frac{14}{3}$. However, we must verify no roots make denominators zero. After checking, all roots are valid, giving sum $-\frac{11}{3}$.
Correct Answer: 2