\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{amsmath}
\usepackage[left=2cm,right=2cm,top=1cm,bottom=2cm]{geometry}
\usepackage{tikz}
	\usetikzlibrary{calc,decorations.pathreplacing}
\usepackage{fancyhdr}
	\fancyfoot[C]{mathweb.fr}
	\renewcommand{\headrulewidth}{0pt}
	
\tikzstyle{titre} = [ 
	rounded corners = 25pt,%
	line width = 0pt,%
	draw,
	color = red!50!black ,%
	shading = axis,%
	rectangle,%
	left color = red!50!black, 
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	shading angle = 135,%
	anchor=north,%
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	align = center
	]
\newcommand{\lmss}{\fontfamily{lmss}\selectfont}
\begin{document}
\thispagestyle{fancy}
\begin{center}
\begin{tikzpicture}
\node[titre] {\textbf{Fiche : développements, factorisations et équations}};
\end{tikzpicture}
\end{center}

\paragraph*{\lmss Développements}

\begin{center}
\begin{tikzpicture}[scale=2]
\node[red] (fact) {$(3x+1)(5x-2)$};
\node[right] (egal) at (fact.east) {$=$};
\node[right,blue] (dev) at (egal.east) {$15x^2-6x+5x-2$};
\node[right] (eg) at (dev.east) {$=$};
\node[right,purple] (reduite) at (eg.east) {$15x^2-x-2$};
%
\draw[->,>=latex,orange,dashed] (fact.160) to[bend left=40] (fact.20);
\node[orange] at ($(fact.north)+(0,10pt)$) {$\times$};
\draw[->,>=latex,orange] (fact.230) to[bend right=20] (fact.310);
\draw[->,>=latex,orange,dashed] (fact.230) to[bend right=40] (fact.340);
%
\draw [red, decorate, decoration = { brace, amplitude = 10pt}, xshift=0pt, yshift=-2pt] ($(fact.south east)+(0,-5pt)$) -- ($(fact.south west)+(0,-5pt)$);
\draw [blue, decorate, decoration = { brace, amplitude = 10pt}, xshift=0pt, yshift=-2pt] ($(dev.south east)+(0,-5pt)$) -- ($(dev.south west)+(0,-5pt)$);
\draw [purple, decorate, decoration = { brace, amplitude = 10pt}, xshift=0pt, yshift=-2pt] ($(reduite.south east)+(0,-5pt)$) -- ($(reduite.south west)+(0,-5pt)$);
%
\draw[->,>=latex,orange] (fact.160) to[bend left=20] (fact.60);
%
\node[red,below] at ($(fact.south)+(0,-15pt)$) {forme factorisée};
\node[blue,below] at ($(dev.south)+(0,-15pt)$) {forme développée};
\node[purple,text width = 4cm,align=center,below] at ($(reduite.south)+(0,-15pt)$) {forme développée réduite};
%
\draw[thick,green!50!black] (dev.200) -- (dev.342);
\draw[thick,green!50!black] (reduite.240) -- (reduite.320);
\draw[green!50!black,->,>=latex] (dev.271) -- ($(dev.271)+(0,-3pt)$) -- ($(reduite.280)+(0,-3pt)$) -- (reduite.280);
\end{tikzpicture}
\end{center}

\paragraph*{\lmss Factorisations}

\begin{center}
\begin{tikzpicture}
\node (fact) {$\textcolor{red}{(2x+1)}(3x-5) - 2(8x+7)\textcolor{red}{(2x+1)}$};
\node[right] (egal) at (fact.east) {$=$};
\node[right] (dev) at (egal.east) {$\textcolor{red}{(2x+1)}\big[ (3x-5)-2(8x+7)\big]$};
\draw[red,<->,>=latex] (fact.188) -- ($(fact.188)+(0,-10pt)$) -- ($(fact.352)+(0,-10pt)$) -- (fact.352);
\node[below,red,text width = 3cm] (factcom) at ($0.5*(fact.188)+0.5*(fact.352)+(0,-10pt)$) {facteur commun des deux blocs};
\draw[->,>=latex,red,dashed] (factcom.east) to[bend right=20] (dev.192);
%
\draw [blue, decorate, decoration = { brace, amplitude = 10pt}, xshift=0pt, yshift=-2pt] ($(fact.175)+(0,5pt)$) -- ($(fact.140)+(0,5pt)$);
\draw [blue, decorate, decoration = { brace, amplitude = 10pt}, xshift=0pt, yshift=-2pt] ($(fact.90)+(0,5pt)$) -- ($(fact.6)+(0,5pt)$);
%
\node[blue] at ($(fact.170)+(0,25pt)$) {bloc 1};
\node[blue] at ($(fact.12)+(0,25pt)$) {bloc 2};
%
\node[right] (line2) at ($(egal.south west)+(0,-2)$) {$=(2x+1)(3x-5-16x-14)$};
\node[below right] at (line2.south west) {$=\underline{(2x+1)(-13x-19)}$};
%
\draw[purple,->,>=latex] (dev.280) -- ($(dev.280)+(0,-1.5)$);
\node[right,purple,text width=4cm] at ($(dev.280)+(0,-0.7)$) {\small on enlève les\\ parenthèses dans les\\ crochets};
\end{tikzpicture}
\end{center}

\paragraph*{\lmss Identités remarquables}

\begin{center}
\begin{tikzpicture}
\node (tab) { $\begin{array}{r@{~=~}l}
\color{red}(a+b)^2 & \color{blue} a^2 + b^2 +2ab\\[8pt]
\color{red}(a-b)^2 & \color{blue} a^2 + b^2 -2ab\\[8pt]
\color{red}(a+b)(a-b) & \color{blue} a^2-b^2
\end{array}$ };
\draw[->,>=latex] (tab.110) to[bend left=30] (tab.60);
\draw[->,>=latex] (tab.300) to[bend left=30] (tab.240);
\node[above] at ($(tab.north)+(0,5pt)$) {\color{blue}développer};
\node[below] at ($(tab.south)+(0,-5pt)$) {\color{red}factoriser};
\end{tikzpicture}
\end{center}


\paragraph*{\lmss \'Equations au produit nul (exemple)}

\begin{center}
\begin{tikzpicture}
\node (eq) {$(2x+7)(4x-3)=0$};
\node[below] (sol1) at ($(eq.west)+(0,-1)$) { $\begin{aligned} 2x+7 & =0\\2x&=-7\\x&=-\dfrac{7}{2} \end{aligned}$ };
\node[below] (sol2) at ($(eq.east)+(0,-1)$) { $\begin{aligned} 4x-3 & =0\\4x&=3\\x&=\dfrac{4}{3} \end{aligned}$ };
\draw[->,>=latex] (eq.200) -- (sol1.north);
\draw[->,>=latex] (eq.315) -- (sol2.north);
\end{tikzpicture}
\end{center}
\end{document}