jmc

Since $p(x)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5,$ then it must be factor of $$3(x^4+6x^2+25)-(3x^4+4x^2+28x+5)=14x^2-28x+70=14(x^2-2x+5).$$Hence, $p(x)=x^2-2x+5,$ and $p(1)=1-2+5=\boxed{4}.$