[tex]f(x)= e^x (ax^2+bx+c) \\ \\ f(0) = 0 \Rightarrow 1\cdot (0+0+c) = 0 \Rightarrow \boxed{c = 0}\\ \\ f'(x) = e^x(2ax+b)+e^x(ax^2+bx) \\ f'(0) =1 \Rightarrow 1\cdot (0+b)+1\cdot (0+0) = 1 \Rightarrow\boxed{b = 1} \\ \\ f'(x) = e^x(2ax+1+ax^2+x) = e^x(2ax+ax^2+x+1)\\ f''(x) = e^x(2a+2ax+1)+e^x(2ax+ax^2+x+1)\\f''(0) = 4 \Rightarrow 2a+1+1 = 4 \Rightarrow 2a=2 \Rightarrow \boxed{a = 1}[/tex]
