Given,

$A=\{-4,-3,-2,0,1,3,4\}$ and $R=\left\{\right(a,b)\in A\times A$ : $b=\left|a\right|$ or $\left.b^2=a+1\right\}$ be a relation on $A$,

So, the relation is given by,

$R=\left\{\right(-4,4),(-3,3),(0,0),(1,1),(3,3),(4,4),(0,1),(3,-2\left)\right\}$

Now, relation to be reflexive $(a,a)\in R \forall a\in A$

$\Rightarrow (-4,-4),(-3,-3),(-2,-2)$ also should be added in $R$.

Now relation to be symmetric if $(a,b)\in R$, then $(b,a)\in R \forall a,b\in A$

$\Rightarrow (4,-4),(3,-3),(1,0),(-2,3)$ also should be added in $R$

Hence, minimum number of elements to be added to
$R=3+4=7$