2x^3+x^2-8x+3>0

let f(x) = 2x^3+x^2-8x+3

f(1) ? 0
f(-1) = -2+1+8+3 ? 0
f(3) ? 0
f(-3) ? 0
f(3/2) = 0 , yeahhhh, 2x-3 is a factor

by long division,
2x^3+x^2-8x+3 = (2x-3)(x^2 + 2x -1)
the quadratic has roots of
x = -1 ± ?2

looking at the graph of y = 2x^3+x^2-8x+3

-1-?2 < x < -1+?2 or x > 3/2

http://www.wolframalpha.com/input/?i=plot+y+%3D+2x%5E3%2Bx%5E2-8x%2B3