Question

Find ${}^{\prime }{p}^{\prime }$ for which $\left(p-2\right)x^2+2\left(p+1\right)x+p+4<3$ $\forall$ $x$

Answer 1

$(p-2)x^2+2(p+1)x+(p+4)<3$

$\Rightarrow (p-2)x^2+2(p+1)x+p+4-3<0$ and $p-2\ne 0$

$\Rightarrow (p-2)x^2+2(p+1)x+p+1<0$ and $p\ne 2$

Discriminant$=4{(p+1)}^2-4\times (p-2)(p+1)<0$ and $p\ne 2$

$\Rightarrow (p+1)(p+1-p+2)<0$ and $p\ne 2$

$\Rightarrow 3(p+1)<0$ and $p\ne 2$

$\Rightarrow p\in (-\infty ,-1)$