1. **State the problem:** Solve the equation $5kx^2 + 6x = 2(x+4)$ for $x$. 2. **Rewrite the equation:** Expand the right side: $$5kx^2 + 6x = 2x + 8$$ 3. **Bring all terms to one side:** $$5kx^2 + 6x - 2x - 8 = 0$$ $$5kx^2 + 4x - 8 = 0$$ 4. **Identify the quadratic form:** This is a quadratic equation in $x$: $$ax^2 + bx + c = 0$$ where $a = 5k$, $b = 4$, and $c = -8$. 5. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 6. **Substitute values:** $$x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 5k \cdot (-8)}}{2 \cdot 5k}$$ $$x = \frac{-4 \pm \sqrt{16 + 160k}}{10k}$$ 7. **Simplify the denominator by canceling common factors:** $$x = \frac{\cancel{-4} \pm \sqrt{16 + 160k}}{\cancel{10k}}$$ (Note: No common factor to cancel here, so this step is just to show the process.) 8. **Final solution:** $$\boxed{x = \frac{-4 \pm \sqrt{16 + 160k}}{10k}}$$ This gives the two possible values of $x$ depending on the sign chosen and the value of $k$. **Important:** The discriminant $16 + 160k$ must be non-negative for real solutions, so $k \geq -\frac{1}{10}$.