We want to write [tex]x^2+8x-3[/tex] in the form [tex](x-a)^2+b[/tex]. To do that, if we expand the second expression and set it equal to the first, we get
[tex]x^2-2ax+a^2+b=x^2+8x-3[/tex]
So we need to have
[tex]\begin{cases}-2a=8\\a^2+b=-3\end{cases}[/tex]
The first condition tells us that [tex]a=-4[/tex], so [tex](-4)^2+b=-3\implies b=-19[/tex]. Then
[tex]x^2+8x-3=(x+4)^2-19=0\implies(x+4)^2=19\implies x+4=\pm\sqrt{19}[/tex]
[tex]\implies x=-4\pm\sqrt{19}[/tex]
