The number of distinct real roots of the

equation

                  $(x+3{)}^4+(x+5{)}^4=16 \left(1\right)$

is

Put $x + 4 = t,$ so that $\left(1\right)$ becomes 

$\begin{array}{l} (t-1{)}^4+(t+1{)}^4=16\\ \Rightarrow 2\left(t^4+6t^2+1\right)=16\\ \Rightarrow t^4+6t^2-7=0\\ \Rightarrow t^2=1,-7\Rightarrow t=\pm 1,\sqrt{7}i\end{array}$

Thus, the equation (1) has two real roots.