I think you meant 2x3+x2+8x+4 because the given polynomial doesn't factor.
2x3+x2+8x+4=x2(2x+1)+4(2x+1)=(x2+4)(2x+1).
Synthetic division confirms that the real zero is x=-½.
-½ | 2 1 8 4
2 -1 0 | -4
2 0 8 | 0 = 2x2+8=2(x2+4).
2(x2+4)(x+½)=(x2+4)(2x+1).