Question 583247
How to find the end behavior of this function? 
x(2-x)(x+3)^2 
**
zeros are: 0, 2 and -3 (multi 2)
Draw a number line with the zeros on it to show how the sign of f(x) changes as we go thru the zeros.

<..-...-3....-.....0.....+....2....-....>

When x is >2, f(x)<0
Moving to the left, signs of the intervals switch as we go thru zeros of odd multiplicity like 1, 3, 5 etc. but do not switch when going thru zeros of even multiplicity like 2, 4, 6 etc. For this problem, 3 is of even multiplicity 2, while 0 and 2 are of odd multiplicity 1.
You are right that as x approaches negative or positive infinity, f(x) approaches negative infinity.