1. **State the problem:** Solve for $x$ in the equation $$\sqrt[3]{5x + 7} = 4.$$\n\n2. **Formula and rules:** To solve an equation involving a cube root, we can cube both sides to eliminate the cube root because cubing is the inverse operation of taking the cube root.\n\n3. **Step 1: Cube both sides** to remove the cube root:\n$$\left(\sqrt[3]{5x + 7}\right)^3 = 4^3$$\n$$5x + 7 = 64$$\n\n4. **Step 2: Solve for $x$:**\nSubtract 7 from both sides:\n$$5x + 7 - 7 = 64 - 7$$\n$$5x = 57$$\n\n5. **Step 3: Divide both sides by 5:**\n$$x = \frac{57}{5}$$\nShow cancellation for clarity:\n$$x = \frac{\cancel{57}}{\cancel{5}}$$ (no common factors to cancel, so fraction remains)\n\n6. **Final answer:**\n$$x = \frac{57}{5} = 11.4.$$