If $x^2+2px+5p-6>0,x\in R$, then

$f\left(x\right)=x^2+2px+5p-6>0$

$a=1>0,Exp>0$ when $\Delta <0$

$b^2-4ac<0$

$\Rightarrow {\left(2p\right)}^2-4\left(1\right)(5p-6)<0$

$\Rightarrow 4p^2-4(5p-6)<0$

$\Rightarrow p^2-5p+6<0$

$(p-2)(p-3)<0$

$2<p<3$