math ergent please please...
posted by kavi on .
10.a)Solve the inequality x^35x^2+2x+8<0 by
i) using intervals
ii) considering all cases
I did this but my teacher said that when i solved this: x^35x^2+2x+8<0
(x+1)(x^26x+8)
(x+1)(x4)(x2)<0
She told me to show all the steps, but i don't know how to show more. so can you please help me to show all the works and rest of the works i did too but she said " the same answer has to happen for each method. i don't really understand that. can you help me please and thank you....
Steve explained this, but still i didn't got the steps. can someone do it for me step by step so i can understand the steps of it please.

Ok, so all the algebra work has been done
x^35x^2+2x+8<0
(x+1)(x4)(x2)<0
Quickest way:
I would look at the graph of
y = x^35x^2+2x+8
= (x+1)(x4)(x2)
I know that this is standard cubic function opening up in quadrant I and having xintercepts of 1, 2, and 4
So a very rough sketch shows me that this curve is below the xaxis ( < 0 ) for x <  1 or 2 < x < 4
looking at all cases , the critical values of 2,2, and 4 split the line into 4 parts :
if x < 1, try some value, e.g. x = 5
()()(+) < 0 , so yes
x < 1
if x is between 2 and 2 , e.g. x = 0
(+)()(2) > 0 , so NO
if x is between 2 and 4 , e.g. x = 3
(+)()(+) < 0 , so YES
x is between 2 and 4
if x > 4 , e.g. x = 5
(+)(+)(+) > 0 , NO
then x < 2 OR 2 < x < 4 
Reiny help me with using intervals. u forgot to help me with that. its urgent please