Step-by-step Solution:

The given equation is: \[ 5^{1 + |\sin x| + |\sin x|^2 + \ldots} = 25 \] Step 1: Simplify the Exponent The exponent is an infinite geometric series: \[ 1 + |\sin x| + |\sin x|^2 + \ldots \] For a geometric series with first term \(a = 1\) and common ratio \(r = |\sin x|\), the sum is: \[ \text{Sum} = \frac{1}{1 - r} = \frac{1}{1 - |\sin x|} \] provided \( |\sin x| < 1 \). \[\] Step 2: Rewrite the Equation Substitute the sum of the geometric series into the equation: \[ 5^{\frac{1}{1 - |\sin x|}} = 25 \] Since \(25 = 5^2\), equate the exponents: \[ \frac{1}{1 - |\sin x|} = 2 \] Step 3: Solve for \(|\sin x|\) Rearrange the equation: \[ 1 - |\sin x| = \frac{1}{2} \] \[ |\sin x| = \frac{1}{2} \] Step 4: Find the Number of Solutions for \(x\) The condition \(|\sin x| = \frac{1}{2}\) implies: \[ \sin x = \pm \frac{1}{2} \] Within the interval \(x \in (-\pi, \pi)\), the solutions for \(\sin x = \frac{1}{2}\) are: \[ x = \frac{\pi}{6}, \quad x = \frac{5\pi}{6} \] And for \(\sin x = -\frac{1}{2}\), the solutions are: \[ x = -\frac{\pi}{6}, \quad x = -\frac{5\pi}{6} \] Thus, there are a total of \(4\) solutions.