To determine the values of \(x\), given the set of numbers \(59, 53, 51, 43, 36, (8x + 1), (x^2 + 1), 13, 12, 9, 7, 3\) with a median of 25, follow these steps:

1. First, note that there are 12 numbers in total. As the median is the middle value(s), for 12 numbers, the median will be the average of the 6th and 7th numbers when they are in descending order.
2. Arrange the non-variable numbers in descending order: \(59, 53, 51, 43, 36, 13, 12, 9, 7, 3\).
3. Insert the expressions involving \(x\): \(8x + 1\) and \(x^2 + 1\) into the correct positions in the list. We know these fall around the median, so they must be the 6th and 7th, or adjacent.
4. Given the median is 25, it follows that: \(\frac{(8x + 1) + (x^2 + 1)}{2} = 25\).
5. Solving this equation: 

1. This is a quadratic equation: \(x^2 + 8x - 48 = 0\). Solve it using factorization or the quadratic formula:
2. Try factorization: 

1. Evaluate each solution:
   • If \(x = 4\), then \(8x + 1 = 33\) and \(x^2 + 1 = 17\). Insert these numbers into our sequence: \(59, 53, 51, 43, 36, 33, 17, 13, 12, 9, 7, 3\). The median is indeed 25 (average of 17 and 33).
   • If \(x = -12\), then \(8x + 1 = -95\) and \(x^2 + 1 = 145\). Insert these numbers: \(145, 59, 53, 51, 43, 36, 13, 12, 9, 7, 3, -95\). Here, the median is not 25, so \(x = -12\) is invalid.
2. Only \(x = 4\) ensures the median is 25, confirming the correct answer is \(4 \text{ only}\).