1. The problem is to find the derivative $\frac{dy}{dx}$ of the implicit function given by the equation $$y^3 + y^2 - 5y - x^2 = -4.$$ 2. We use implicit differentiation. Differentiate both sides with respect to $x$. Remember that $y$ is a function of $x$, so when differentiating terms with $y$, apply the chain rule: $$\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}.$$ 3. Differentiate each term: - $\frac{d}{dx}[y^3] = 3y^2 \frac{dy}{dx}$ - $\frac{d}{dx}[y^2] = 2y \frac{dy}{dx}$ - $\frac{d}{dx}[-5y] = -5 \frac{dy}{dx}$ - $\frac{d}{dx}[-x^2] = -2x$ - $\frac{d}{dx}[-4] = 0$ 4. Substitute these into the differentiated equation: $$3y^2 \frac{dy}{dx} + 2y \frac{dy}{dx} - 5 \frac{dy}{dx} - 2x = 0.$$ 5. Group terms with $\frac{dy}{dx}$: $$\left(3y^2 + 2y - 5\right) \frac{dy}{dx} = 2x.$$ 6. Solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{2x}{3y^2 + 2y - 5}.$$ 7. This matches option B. Final answer: $$\boxed{\frac{dy}{dx} = \frac{2x}{3y^2 + 2y - 5}}.$$