\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}
  
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{arrows, positioning, calc, intersections}
 
\begin{document}

% Define this as a command to ensure that it is same in both cases
\newcommand*{\ShowIntersection}[2]{
\fill 
    [name intersections={of=#1 and #2, name=i, total=\t}] 
    [red, opacity=1, every node/.style={above left, black, opacity=1}] 
    \foreach \s in {1,...,\t}{(i-\s) circle (2pt)
        node [above left] {\s}};
}

\begin{preview}
\begin{tikzpicture}
    \begin{axis}[
        axis x line=middle,
        axis y line=middle,
        width=15cm, height=15cm,     % size of the image
        grid = major,
        grid style={dashed, gray!30},
        %xmode=log,log basis x=10,
        %ymode=log,log basis y=10,
        xmin=-2,     % start the diagram at this x-coordinate
        xmax= 4,    % end   the diagram at this x-coordinate
        ymin=-7,     % start the diagram at this y-coordinate
        ymax= 7,   % end   the diagram at this y-coordinate
        %/pgfplots/xtick={0,1,...,60}, % make steps of length 5
        %extra x ticks={23},
        %extra y ticks={0.507297},
        axis background/.style={fill=white},
        ylabel=y,
        xlabel=x,
        %xticklabels={,,},
        %yticklabels={,,},
        tick align=outside,
        tension=0.08]
      % plot the stirling-formulae
      \addplot[name path global=a, domain=-2:4, red, thick,samples=500] 
        {-x*x*x + 4*x*x-x-4}; 
      \addplot[name path global=b, domain=0.25:4, blue, thick, dashed] {x-1.2}; 
      \addplot[name path global=c, domain=0.25:4, blue, thick, dashed] {x+0.42}; 
      \ShowIntersection{a}{b};
    \end{axis} 
\end{tikzpicture}
\end{preview}
\end{document}