To find $\frac{dy}{dx}$ using implicit differentiation, let's start with the equation:

$x^3(x + y) = y^2(4x - y)$

Step 1: Differentiate both sides with respect to $x$

Using the product rule, differentiate each side:

Left Side

$\frac{d}{dx} \left( x^3(x + y) \right) = x^3 \cdot \frac{d}{dx}(x + y) + (x + y) \cdot \frac{d}{dx}(x^3)$ The derivatives are: $= x^3(1 + \frac{dy}{dx}) + (x + y) \cdot 3x^2$ Expanding: $= x^3 + x^3 \frac{dy}{dx} + 3x^2(x + y)$

Right Side

$\frac{d}{dx} \left( y^2(4x - y) \right) = y^2 \cdot \frac{d}{dx}(4x - y) + (4x - y) \cdot \frac{d}{dx}(y^2)$ The derivatives are: $= y^2(4 - \frac{dy}{dx}) + (4x - y) \cdot 2y \frac{dy}{dx}$ Expanding: $= 4y^2 - y^2 \frac{dy}{dx} + 8xy \frac{dy}{dx} - 2y^2 \frac{dy}{dx}$

Step 2: Combine Terms

Now, we have:

$x^3 + x^3 \frac{dy}{dx} + 3x^3 + 3x^2y = 4y^2 - y^2 \frac{dy}{dx} + 8xy \frac{dy}{dx} - 2y^2 \frac{dy}{dx}$

Step 3: Solve for $\frac{dy}{dx}$

Combine