This equation keep giving me error messages

Your problems is already solved in other answers, but you may liked the following solution which employ the instruction MoveEqLeft defined in the mathtools package:

documentclass{article}
usepackage{mathtools}

begin{document}
begin{align*}
    MoveEqLeft[3]
frac{6sqrt{{(x+1)(x+3)+1}-12}}{2x}        \
     & =frac{6sqrt{x^2+4x+3+1}-12}{2x}     
       =frac{6sqrt{x^2+4x+4}-12}{2x}      \ & =frac{6(x+2)-12}{2x}                 
       =frac{6x+12-12}{2x}                 \
     & =frac{6x}{2x}=3
end{align*}
end{document}

(red lines indicate text area borders)

Errors in the first line are the cause, ie mismatched { }. A fix is:

begin{align*}
frac{6sqrt{(x+1)(x+3)+1}-12}{2x}
&=frac{6sqrt{x^2+4x+3+1}-12}{2x} \
&=frac{6sqrt{x^2+4x+4}-12}{2x} \
&=frac{6(x+2)-12}{2x} \
&=frac{6x+12-12}{2x} \
&=frac{6x}{2x} = 3
end{align*}

which produces:

Note

• align* environment enters math mode for you, so there is node need to wrap it in between $$ ... $$.
• The left brace of numerator of first frac is never closed by a pairing right brace.
• You might need & mathrel{phantom{=}} prepended to first line of align*.

documentclass{article}
usepackage{amsmath}

begin{document}
begin{align*}
     & mathrel{phantom{=}}frac{6sqrt{{(x+1)(x+3)+1}-12}}{2x} \
     & =frac{6sqrt{x^2+4x+3+1}-12}{2x}                         \
     & =frac{6sqrt{x^2+4x+4}-12}{2x}                           \
     & =frac{6(x+2)-12}{2x}                                     \
     & =frac{6x+12-12}{2x}                                      \
     & =frac{6x}{2x}=3
end{align*}
end{document}