Let $b$ be a real number randomly sepected from the interval $[-17,17]$.  Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions.  Find the value of $m+n$.
1. Start by rearranging the given equation:
   \[
   x^4 + 25b^2 = (4b^2 - 10b)x^2
   \]
   This can be rewritten as:
   \[
   x^4 - (4b^2 - 10b)x^2 + 25b^2 = 0
   \]
   Let \( y = x^2 \). Then the equation becomes a quadratic in \( y \):
   \[
   y^2 - (4b^2 - 10b)y + 25b^2 = 0
   \]

2. For the quadratic equation \( y^2 - (4b^2 - 10b)y + 25b^2 = 0 \) to have at least two distinct real solutions, its discriminant must be non-negative:
   \[
   \Delta = (4b^2 - 10b)^2 - 4 \cdot 1 \cdot 25b^2
   \]
   Simplify the discriminant:
   \[
   \Delta = (4b^2 - 10b)^2 - 100b^2
   \]
   \[
   \Delta = 16b^4 - 80b^3 + 100b^2 - 100b^2
   \]
   \[
   \Delta = 16b^4 - 80b^3
   \]
   Factor out the common term:
   \[
   \Delta = 16b^3(b - 5)
   \]

3. For the discriminant to be positive:
   \[
   16b^3(b - 5) > 0
   \]
   This inequality holds when \( b \) is in the intervals:
   \[
   b < 0 \quad \text{or} \quad b > 5
   \]

4. Determine the probability that \( b \) falls within these intervals. The interval for \( b \) is \([-17, 17]\). The intervals where the discriminant is positive are \( b < 0 \) and \( b > 5 \).

5. Calculate the lengths of these intervals:
   - For \( b < 0 \), the interval is \([-17, 0)\), which has a length of 17.
   - For \( b > 5 \), the interval is \((5, 17]\), which has a length of \(17 - 5 = 12\).

6. The total length of the interval where the discriminant is positive is:
   \[
   17 + 12 = 29
   \]

7. The total length of the interval \([-17, 17]\) is 34. Therefore, the probability that \( b \) falls within the desired intervals is:
   \[
   \frac{29}{34}
   \]

8. Since \( m \) and \( n \) are relatively prime positive integers such that \( \frac{m}{n} = \frac{29}{34} \), we have \( m = 29 \) and \( n = 34 \).

9. The value of \( m + n \) is:
   \[
   29 + 34 = 63
   \]

The final answer is \( \boxed{63} \).