We know that $(a+b)^2=|\;a\;|^2+|\;b\;|^2+2 \;\overrightarrow a.\;\overrightarrow b$
$\qquad\qquad\qquad=|\;a\;|^2+|\;b\;|^2+ 2 |\;a\;||\;b\;| \cos \theta$
But it is given that $(a+b)^2=|\;a\;|^2+|\;b\;|^2+2 |\overrightarrow a||\;\overrightarrow b\;|$
This implies $\cos \theta=1 (ie) \theta=0$
This implies that the angle between the vectors is zero. That is they are parallel vector.
Hence $|\;a\;|^2+|\;b\;|^2+2 |\overrightarrow a||\;\overrightarrow b\;|$ is valid for non zero vectors. $\overrightarrow a$ and $\overrightarrow b$ only if they are parallel
Hence the statment is $False$