\[\begin{array}{l}
y = a{x^2} + x + c\,\,\,\,\left( P \right)\\
b)\,\,\,\left( P \right)\,\,\,co\,\,\,truc\,\,doi\,\,xung\,\,la\,\,\,x = - \frac{3}{2}\\
\Rightarrow - \frac{1}{{2a}} = - \frac{3}{2} \Leftrightarrow a = \frac{1}{3} \Rightarrow \left( P \right):\,\,\,y = \frac{1}{3}{x^2} + x + c.\\
\left( P \right)\,\,\,di\,\,\,qua\,\,diem\,\,\,\left( {3; - 4} \right)\\
\Rightarrow - 4 = \frac{1}{3}{.3^2} + 3 + c \Leftrightarrow c = - 10.\\
Vay\,\,\,a = \frac{1}{3};\,\,c = - 10.\\
c)\,\,\,I\left( {2;\,\, - 2} \right)\\
\left( P \right)\,\,\,co\,\,\,dinh\,\,\,I\left( {2;\,\, - 2} \right)\\
\Rightarrow \left\{ \begin{array}{l}
- \frac{1}{{2a}} = 2\\
a{.2^2} + 2 + c = - 2
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = - \frac{1}{4}\\
c = - 5
\end{array} \right.\\
Vay\,\,\,a = - \frac{1}{4};\,\,\,c = - 5.\\
d)\,\,\,\left( P \right)\,\,\,\,cat\,\,\,Oy\,\,\,tai\,\,\,diem\,\,\,co\,\,tung\,\,do\,\,\, = - \frac{1}{4}\\
\Rightarrow \left( P \right)\,\,\,\,di\,\,\,qua\,\,\,C\left( {0; - \frac{1}{4}} \right)\\
\Rightarrow c = - \frac{1}{4}.\\
\Rightarrow \left( P \right):\,\,\,y = a{x^2} + x - \frac{1}{4}\\
\left( P \right)\,\,\,\,di\,\,\,qua\,\,\,\,D\left( { - 1;\,\,6} \right)\\
\Rightarrow 6 = a.1 - 1 - \frac{1}{4}\\
\Rightarrow a = \frac{{29}}{4}.\\
Vay\,\,\,\,a = \frac{{29}}{4},\,\,\,c = - \frac{1}{4}.
\end{array}\]
