1. **State the problem:** Solve the equation $$(x+1)(x+3)(x+5)(x+7) = 9$$ for $x$. 2. **Use a substitution to simplify:** Notice the terms are symmetric around the midpoint between 4 and 5. Let $$y = x + 4$$. Then the factors become: $$(x+1) = y - 3, \quad (x+3) = y - 1, \quad (x+5) = y + 1, \quad (x+7) = y + 3$$ 3. **Rewrite the product:** $$ (y-3)(y-1)(y+1)(y+3) = 9 $$ Group as: $$ [(y-3)(y+3)] \times [(y-1)(y+1)] = 9 $$ 4. **Use difference of squares:** $$ (y^2 - 9)(y^2 - 1) = 9 $$ 5. **Expand the product:** $$ y^4 - y^2 - 9y^2 + 9 = 9 $$ Simplify: $$ y^4 - 10y^2 + 9 = 9 $$ 6. **Subtract 9 from both sides:** $$ y^4 - 10y^2 + 9 - 9 = 0 $$ $$ y^4 - 10y^2 = 0 $$ 7. **Factor out $y^2$:** $$ y^2(y^2 - 10) = 0 $$ 8. **Solve each factor:** - Case 1: $$ y^2 = 0 \implies y = 0 $$ - Case 2: $$ y^2 - 10 = 0 \implies y^2 = 10 \implies y = \pm \sqrt{10} $$ 9. **Recall substitution $y = x + 4$ and solve for $x$:** - For $$y=0$$: $$x + 4 = 0 \implies x = -4$$ - For $$y = \sqrt{10}$$: $$x + 4 = \sqrt{10} \implies x = -4 + \sqrt{10}$$ - For $$y = -\sqrt{10}$$: $$x + 4 = -\sqrt{10} \implies x = -4 - \sqrt{10}$$ 10. **Final solutions:** $$ x = -4, \quad x = -4 + \sqrt{10}, \quad x = -4 - \sqrt{10} $$