1. **State the problem:** Solve the equation $$(x+1)(x+3)(x+5)(x+7) = 9.$$\n\n2. **Rewrite the equation:** Notice the factors are symmetric around the midpoint between 4 and 5. Group them as $$(x+1)(x+7) \text{ and } (x+3)(x+5).$$\n\n3. **Simplify each group:**\n$$(x+1)(x+7) = x^2 + 8x + 7,$$\n$$(x+3)(x+5) = x^2 + 8x + 15.$$\n\n4. **Substitute:** Let $$y = x^2 + 8x.$$ Then the product becomes $$(y + 7)(y + 15) = 9.$$\n\n5. **Expand and form quadratic in y:**\n$$y^2 + 22y + 105 = 9,$$\nwhich simplifies to\n$$y^2 + 22y + 96 = 0.$$\n\n6. **Solve quadratic for y:** Use the quadratic formula $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $$a=1, b=22, c=96.$$\nCalculate discriminant:\n$$\Delta = 22^2 - 4 \times 1 \times 96 = 484 - 384 = 100.$$\n\n7. **Find roots for y:**\n$$y = \frac{-22 \pm 10}{2}.$$\nSo, $$y_1 = \frac{-22 + 10}{2} = -6,$$\n$$y_2 = \frac{-22 - 10}{2} = -16.$$\n\n8. **Recall substitution:** $$y = x^2 + 8x.$$ Solve for x in each case.\n\nFor $$y = -6$$:\n$$x^2 + 8x = -6 \implies x^2 + 8x + 6 = 0.$$\nDiscriminant:\n$$64 - 24 = 40.$$\nRoots:\n$$x = \frac{-8 \pm \sqrt{40}}{2} = -4 \pm \sqrt{10}.$$\n\nFor $$y = -16$$:\n$$x^2 + 8x = -16 \implies x^2 + 8x + 16 = 0.$$\nDiscriminant:\n$$64 - 64 = 0.$$\nRoot:\n$$x = \frac{-8}{2} = -4.$$\n\n9. **Final solutions:**\n$$x = -4 + \sqrt{10}, \quad x = -4 - \sqrt{10}, \quad x = -4.$$