1. Can \alpha=\beta ?I thought its stated clearly that

\alpha<-1<\beta
No, they're not equal at all. The only thing we change is 'what' we call $\displaystyle \alpha,\beta$

We call the roots, $\displaystyle \alpha,\beta$ with $\displaystyle \alpha < -1 < \beta$

We simply call $\displaystyle \alpha$, the smallest of the 2 roots, and $\displaystyle \beta$ the other. Thus if the inequality '<' flips into '>' due to a negative sign, then we only change what we call $\displaystyle \alpha,\beta$. Perhaps, I was unclear.

I hope you understand what I mean, now.

2. Originally Posted by Dinkydoe
No, they're not equal at all. The only thing we change is 'what' we call $\displaystyle \alpha,\beta$

We call the roots, $\displaystyle \alpha,\beta$ with $\displaystyle \alpha < -1 < \beta$

We simply call $\displaystyle \alpha$, the smallest of the 2 roots, and $\displaystyle \beta$ the other. Thus if the inequality '<' flips into '>' due to a negative sign, then we only change what we call $\displaystyle \alpha,\beta$. Perhaps, I was unclear.

I hope you understand what I mean, now.
yeah, they are unknowns anyways. No worries Dinky, you've been very helpful and precise in explaining. thanks !

3. Originally Posted by hooke
yeah, they are unknowns anyways. No worries Dinky, you've been very helpful and precise in explaining. thanks !
I would like to do a final checking, is the answer -1<m<4 ?

4. I found something slightly different.

I guess that you also found that

$\displaystyle \Delta(f)>(-m^2+6m+10)^2$

could be simplified to $\displaystyle 12m^3-60m^2+24m+96> 0$

This polynomial can be factored:

$\displaystyle g(m)=12(m+1)(m-4)(m-2)$. Thus we find zero's $\displaystyle m=-1,m=2,m=4$

Then we can argue that, (or see by plotting) that $\displaystyle g(m)> 0$ as $\displaystyle -1<m<2$ or $\displaystyle m>4$

5. it is very easy,but your thinking is worry. wow gold

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