The equation $2x^2 + 2(p + 1) x +p= 0$, where $p$ is real, always has roots that are

The discriminant of a quadratic equation

$ax^2 + bx + c = 0$ is given by $b^2-4ac$

$a=2,b=2(p+1)$ and $c=p$

$[2(p+1)]^2-4(2p)\Rightarrow 4(p+1)^2-8p$

$\Rightarrow 4[(p+1)^2-2p]\Rightarrow 4[(p^2+2p+ 1)-2p]$

$\Rightarrow 4(p^2+1)$

For any real value of $p,4(p^2+1)$ will always be positive as $p^2$ cannot be negative for real $p$.

Hence, the discriminant $b^2-4ac$ will always be positive.

When the discriminant is greater than $0$ or is positive, then the roots of a quadratic equation will be real.