The number of integer solutions of the equation
$$ 2|x|(x^2 + 1) = 5x^2 $$
is:

Question

The number of integer solutions of the equation  
$$ 2|x|(x^2 + 1) = 5x^2 $$  
is:

Accepted Answer

To solve the given equation $$ 2|x|(x^2 + 1) = 5x^2 $$, let $$ |x| = k $$. Substituting $$ k $$ into the equation, we get:  
$$
2k(k^2 + 1) = 5k^2
$$

Simplify the equation:  
$$
2k^3 + 2k = 5k^2
$$  
Rearranging terms:  
$$
2k^3 - 5k^2 + 2k = 0
$$  
Factorizing:  
$$
k(2k^2 - 5k + 2) = 0
$$

The solutions are:  
1. $$ k = 0 $$  
2. Solve $$ 2k^2 - 5k + 2 = 0 $$:  
   Using the quadratic formula, $$ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where $$ a = 2, b = -5, c = 2 $$:  
   $$
   k = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)} = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4}
   $$  
   So, $$ k = \frac{8}{4} = 2 $$ or $$ k = \frac{2}{4} = 0.5 $$.

Thus, $$ k = |x| $$ can take values $$ 0, 0.5, 2 $$.

Since $$ x $$ must be an integer, only $$ |x| = 0 $$ and $$ |x| = 2 $$ are valid. Hence, $$ x = 0, -2, 	ext{ or } 2 $$.

Therefore, there are **3 integer solutions** to the equation.