What is ((x² + 4)/(y² + 4)) (dy/dx) [(x² + 4)(d²y/dx²) − 16y] equal to?

Using the result from the previous problem, (dy/dx)² = 16(y² + 4)/(x² + 4), so (x² + 4)(dy/dx)² = 16(y² + 4). Differentiating both sides with respect to x: 2x(dy/dx)² + (x² + 4)·2(dy/dx)(d²y/dx²) = 32y(dy/dx). Dividing by 2(dy/dx): x(dy/dx) + (x² + 4)(d²y/dx²) = 16y, so (x² + 4)(d²y/dx²) − 16y = −x(dy/dx). Multiplying by ((x² + 4)/(y² + 4))(dy/dx) gives ((x² + 4)/(y² + 4))(dy/dx)·(−x dy/dx) = −x(x² + 4)(dy/dx)²/(y² + 4) = −x·16(y² + 4)/(y² + 4) = −16x.