1. **State the problem:** Solve the quadratic equation $$(5n + 1)^2 + 5(5n + 1) - 6 = 0$$ for $n$. 2. **Rewrite the equation:** Let $x = 5n + 1$. Then the equation becomes: $$x^2 + 5x - 6 = 0$$ 3. **Solve the quadratic in $x$:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=5$, and $c=-6$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4(1)(-6) = 25 + 24 = 49$$ 5. **Find the roots for $x$:** $$x = \frac{-5 \pm \sqrt{49}}{2} = \frac{-5 \pm 7}{2}$$ 6. **Evaluate each root:** - For $x = \frac{-5 + 7}{2} = \frac{2}{2} = 1$ - For $x = \frac{-5 - 7}{2} = \frac{-12}{2} = -6$ 7. **Back-substitute $x = 5n + 1$:** - When $x=1$: $$5n + 1 = 1 \implies 5n = 0 \implies n = 0$$ - When $x=-6$: $$5n + 1 = -6 \implies 5n = -7 \implies n = -\frac{7}{5}$$ **Final answer:** $$n = 0, -\frac{7}{5}$$