Let [tex]P(x)=ax^2+bx+c[/tex]. You have derivatives [tex]P'(x)=2ax+b[/tex] and [tex]P''(x)=2a[/tex]. Then
[tex]\begin{cases}P(2)=4a+2b+c=5\\P'(2)=4a+b=3\\P''(2)=2a=2\end{cases}[/tex]
From the last equation, you have [tex]a=1[/tex]. Plug this into the second equation to get [tex]4+b=3\implies b=-1[/tex]. Plug these into the first to get [tex]4-2+c=5\implies c=3[/tex].
So the polynomial is
[tex]P(x)=x^2-x+3[/tex]
