We need to find (x-a)^2 such that
[tex]\begin{gathered} x^2+8x+4+b=(x+a)^2 \\ a,b\rightarrow\text{ constants} \end{gathered}[/tex]Expanding the right side of the equation,
[tex]\begin{gathered} \Rightarrow x^2+8x+4+b=x^2+2ax+a^2 \\ \Rightarrow8x=2ax,4+b=a^2 \\ \Rightarrow a=\frac{8}{2}=4 \\ \Rightarrow4+b=16 \\ \Rightarrow b=12 \end{gathered}[/tex]Therefore, completing the square,
[tex]x^2+8x+4+12=(x+4)^2[/tex]Add a +12 to complete the square.
