Question 1189459
Graph this:
{{{graph(300,300,-10,10,-50,25,2(x+0.5)(x+4)^3)}}}; {{{graph(300,300,-5,5,-5,5,2(x+0.5)(x+4)^3)}}}
By inspection, x=-4 is a root with multiplicity 3, and so is x=-0.5. Both of those make the parentheses 0, and they cross the x-axis.
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Take the derivative of the function:
f'(x)=2(x+0.5)*3(x+4)^2+(x+4)^3(2)
as x approaches -4 from the either side, the derivative is negative, or the slope negative. 
As x approaches 0.5 from either side, the slope is positive.
There is one turning point where the derivative is 0.
2(x+0.5)*3(x+4)^2+2(x+4)^3=0
divide both sides by (x+4)^2
6x+3+2(x+4)=0
8x+11=0
x=-1.375
y=-31.65
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end behavior negative is (-x)^4 positive oo
and for positive is x^4 positive oo