Question

Find the largest integral value of $m$ for which the quadratic expression
$y=x^{2}+(2 m+4) x+7 m+8$ is positive for all $x \in R$

Answer 1

Given expression $>0$ for all $x$.
$\Leftrightarrow \Delta<0$ as leading coefficient $>0$
$\Leftrightarrow 4(m+2)^{2}-4(7 m+8)<0$
$\Leftrightarrow m^{2}+4 m+4-7 m-8<0$
$\Leftrightarrow m^{2}-3 m-4<0$
$\Leftrightarrow(m-4)(m+1)<0$
$\Leftrightarrow-1< m< 4$
$\Rightarrow$ Largest integral value of $m=3$