1. **State the problem:** We are given the function equation $f(x) + 1 = x^2 + 2x + 1$ and need to find all values of $x$ such that $f(f(f(f(f(x))))) = 16^{16}$. 2. **Rewrite the function:** From $f(x) + 1 = x^2 + 2x + 1$, we get $$f(x) = x^2 + 2x + 1 - 1 = x^2 + 2x.$$ 3. **Simplify the function:** Notice that $$f(x) = x^2 + 2x = x(x + 2).$$ 4. **Analyze the iteration:** We want to find $x$ such that applying $f$ five times yields $16^{16}$: $$f^{(5)}(x) = 16^{16}.$$ 5. **Rewrite $16^{16}$:** Since $16 = 2^4$, $$16^{16} = (2^4)^{16} = 2^{64}.$$ 6. **Check for fixed points or patterns:** Let's check if $f(x)$ can be expressed in a simpler form to analyze iterations. 7. **Complete the square:** $$f(x) = x^2 + 2x = (x+1)^2 - 1.$$ 8. **Define $g(x) = f(x) + 1 = (x+1)^2$:** Then $$f(x) = g(x) - 1,$$ and iterating $f$ is related to iterating $g$ minus 1 at each step. 9. **Express iteration in terms of $g$:** $$f^{(n)}(x) = g^{(n)}(x) - 1,$$ where $g^{(n)}$ is $g$ composed $n$ times. 10. **Find $g^{(n)}(x)$:** Since $g(x) = (x+1)^2$, $$g^{(1)}(x) = (x+1)^2,$$ $$g^{(2)}(x) = (g^{(1)}(x) + 1)^2 = ((x+1)^2 + 1)^2,$$ and so on. 11. **We want:** $$f^{(5)}(x) = 2^{64} \\ ightarrow g^{(5)}(x) - 1 = 2^{64} \\ ightarrow g^{(5)}(x) = 2^{64} + 1.$$ 12. **Define $y_0 = x + 1$ and $y_{n} = g^{(n)}(x) = (y_{n-1})^2$:** - $y_1 = (y_0)^2$ - $y_2 = (y_1 + 1)^2$ but from step 10, actually $g(x) = (x+1)^2$, so $g^{(n)}(x) = (( ext{previous}) + 1)^2$. However, the iteration is: $$g^{(1)}(x) = (x+1)^2 = y_1,$$ $$g^{(2)}(x) = (y_1 + 1)^2,$$ $$g^{(3)}(x) = (g^{(2)}(x) + 1)^2,$$ ... and so forth. 13. **Calculate $g^{(5)}(x)$ stepwise:** - $y_1 = (x+1)^2$ - $y_2 = (y_1 + 1)^2$ - $y_3 = (y_2 + 1)^2$ - $y_4 = (y_3 + 1)^2$ - $y_5 = (y_4 + 1)^2$ We want $$y_5 = 2^{64} + 1.$$ 14. **Work backwards:** - $y_5 = 2^{64} + 1$ - $y_4 + 1 = \\sqrt{y_5} = \\sqrt{2^{64} + 1}$ Since $2^{64} = (2^{32})^2$, and $2^{64} + 1$ is not a perfect square, but for the sake of the problem, we keep it symbolic. - $y_4 = \\sqrt{2^{64} + 1} - 1$ - $y_3 + 1 = \\sqrt{y_4} = \\sqrt{\\sqrt{2^{64} + 1} - 1}$ - $y_3 = \\sqrt{\\sqrt{2^{64} + 1} - 1} - 1$ - $y_2 + 1 = \\sqrt{y_3} = \\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1}$ - $y_2 = \\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1$ - $y_1 + 1 = \\sqrt{y_2} = \\sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1}$ - $y_1 = \\sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1} - 1$ - Recall $y_1 = (x+1)^2$, so $$(x+1)^2 = y_1 = \\sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1} - 1.$$ 15. **Solve for $x$:** $$x + 1 = \pm \sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1} - 1}$$ $$x = -1 \pm \sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1} - 1}.$$ 16. **Final answer:** The two values of $x$ satisfying the equation are $$x = -1 + \sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1} - 1}$$ and $$x = -1 - \sqrt{\\sqrt{\\sqrt{\\sqrt{2^{64} + 1} - 1} - 1} - 1} - 1}.$$