1. **State the problem:** Solve the equation $$3x(x + 1) - 7x(x + 2) = 6$$. 2. **Expand each term:** $$3x(x + 1) = 3x^2 + 3x$$ $$-7x(x + 2) = -7x^2 - 14x$$ 3. **Rewrite the equation with expanded terms:** $$3x^2 + 3x - 7x^2 - 14x = 6$$ 4. **Combine like terms:** $$3x^2 - 7x^2 + 3x - 14x = 6$$ $$-4x^2 - 11x = 6$$ 5. **Bring all terms to one side to set equation to zero:** $$-4x^2 - 11x - 6 = 0$$ 6. **Multiply entire equation by -1 to simplify:** $$\cancel{-}4x^2 - \cancel{-}11x - \cancel{-}6 = 0 \Rightarrow 4x^2 + 11x + 6 = 0$$ 7. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=4$, $b=11$, and $c=6$. 8. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 11^2 - 4 \times 4 \times 6 = 121 - 96 = 25$$ 9. **Calculate the roots:** $$x = \frac{-11 \pm \sqrt{25}}{2 \times 4} = \frac{-11 \pm 5}{8}$$ 10. **Find each solution:** - For $+$ sign: $$x = \frac{-11 + 5}{8} = \frac{-6}{8} = -\frac{3}{4}$$ - For $-$ sign: $$x = \frac{-11 - 5}{8} = \frac{-16}{8} = -2$$ **Final answer:** $$x = -\frac{3}{4} \text{ or } x = -2$$