<s>[INST] Prove that roots of cubic equation are real. The equation is $(x+4)(x+2)(x-3)+(x+3)(x+1)(x-5)=0$.
How do I prove that the all the roots are real and distinct?
 [/INST] Let $f(x)=(x+4)(x+2)(x-3)+(x+3)(x+1)(x-5)$. Then


*

*since $-27=f(-4)<0<6=f(-3)$, there is a root in $(-4,-3)$;

*since $7=f(-2)>0>-12=f(-1)$, there is a root in $(-2,-1)$;

*since $-48=f(3)<0<13=f(4)$, there is a root in $(3,4)$.


Since it is a cubic equation, it can have no more roots. Therefore, all roots are real.
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