The number of solutions to the equation $2\sqrt{1 + \sqrt{1 + (x+1)\sqrt{1 + (x+2)\sqrt{1 + (x+3)(x+5)}}}} = x$ is:
Step-by-Step Solution
Key Concept: Recognizing that $(x+3)(x+5) = (x+4)^2 - 1$ allows the nested radical to simplify to an absolute value.
Notice that $\sqrt{[1+(x+3)](x+5)} = |x+4|$ simplifies the inner expression. Since RHS is always positive, the equation becomes $2(x+1) = x$ by simplifying the LHS, which has a unique solution.
Correct Answer: 1