Integer Solutions of the Equation
- To find the integer solutions of the equation 2 |x| (x^2 + 1) = 5x^2, we first need to simplify the equation and then solve for x.

Simplifying the Equation
- We can simplify the equation by expanding the terms and considering the absolute value function.
- By expanding, we get 2x^3 + 2|x| = 5x^2.
- Since |x| can be positive or negative, we have two cases to consider: x ≥ 0 and x < />

Case 1: x ≥ 0
- For x ≥ 0, the absolute value function simplifies to x.
- Substituting x for |x| in the equation gives us 2x^3 + 2x = 5x^2.
- Rearranging terms, we get 2x^3 - 5x^2 + 2x = 0.
- Factoring out x, we get x(2x^2 - 5x + 2) = 0.
- Solving the quadratic equation 2x^2 - 5x + 2 = 0 gives us x = 1 or x = 0.

Case 2: x < />
- For x < 0,="" the="" absolute="" value="" function="" simplifies="" to="" />
- Substituting -x for |x| in the equation gives us 2x^3 + 2(-x) = 5x^2.
- Simplifying, we get 2x^3 - 2x = 5x^2.
- Rearranging terms, we get 2x^3 - 5x^2 - 2x = 0.
- This cubic equation has one integer solution, x = -1.

Conclusion
- Combining the solutions from both cases, we have three integer solutions: x = 0, x = 1, and x = -1.
- Therefore, the number of integer solutions of the equation 2 |x| (x^2 + 1) = 5x^2 is 3.