$a$ and $b$ are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of $2ab.$

[i]Author: Ray Li[/i]
1. Given the equations:
   \[
   a^4 + a^2b^2 + b^4 = 900
   \]
   \[
   a^2 + ab + b^2 = 45
   \]

2. Divide the first equation by the second equation:
   \[
   \frac{a^4 + a^2b^2 + b^4}{a^2 + ab + b^2} = \frac{900}{45}
   \]
   Simplifying the right-hand side:
   \[
   \frac{900}{45} = 20
   \]
   Therefore:
   \[
   a^2 - ab + b^2 = 20
   \]

3. Now we have two equations:
   \[
   a^2 + ab + b^2 = 45
   \]
   \[
   a^2 - ab + b^2 = 20
   \]

4. Add these two equations:
   \[
   (a^2 + ab + b^2) + (a^2 - ab + b^2) = 45 + 20
   \]
   Simplifying the left-hand side:
   \[
   2a^2 + 2b^2 = 65
   \]
   Therefore:
   \[
   a^2 + b^2 = \frac{65}{2}
   \]

5. Substitute \(a^2 + b^2 = \frac{65}{2}\) into the first equation:
   \[
   a^2 + ab + b^2 = 45
   \]
   \[
   \frac{65}{2} + ab = 45
   \]
   Solving for \(ab\):
   \[
   ab = 45 - \frac{65}{2}
   \]
   \[
   ab = 45 - 32.5
   \]
   \[
   ab = 12.5
   \]

6. Finally, calculate \(2ab\):
   \[
   2ab = 2 \times 12.5 = 25
   \]

The final answer is \(\boxed{25}\)