To solve this polynomial equation, we can use various methods such as factoring, the rational root theorem, or synthetic division.

First, let's check for any rational roots using the rational root theorem. The possible rational roots of the polynomial are the factors of the constant term (2) divided by the factors of the leading coefficient (2).

Possible rational roots: ±1, ±2

We can try synthetic division with these possible roots to see if we can find any roots.

Let's try x = 1:

1 | 2 5 -13 -13 5 21 | 2 7 -6 -19 -14 -12

Since the result is not 0, x = 1 is not a root.

Now, let's try x = -1:

-1 | 2 5 -13 -13 5 2-1 | 2 3 10 -3 8 -6

Since the result is not 0, x = -1 is not a root.

Next, let's try x = 2:

2 | 2 5 -13 -13 5 22 | 2 9 5 -21 -37 -72

Since the result is not 0, x = 2 is not a root.

Finally, let's try x = -2:

-2 | 2 5 -13 -13 5 2-2 | 2 1 15 -17 39 -76

Since the result is not 0, x = -2 is not a root.

Since none of the possible rational roots yielded a result of 0, the polynomial likely has no rational roots. We can then attempt to factor the polynomial further or use numerical methods to find approximate solutions.