Synthetic division by root x=-2: 2x^2+3x-2=(2x-1)(x+2). Factors: P(x)=((x+2)^2)(2x-1) Solution: x=-2 (multiple), x=1/2. These are the zeroes. Note that the graph touches the x axis at x=-2 and crosses it at x=1/2.
-2 | 2..7..4 -4
......2 -4 -6..4
......2..3 -2 | 0
y intercept is -4 (when x=0). Degree of polynomial is 3. For x<-2, as x becomes more negative the graph heads towards minus infinity. For x>1/2 the graph heads towards plus infinity as x gets larger.