Using **ComplexExpand** command:

 A = 5 t^2 + 3 t + 4;
 ComplexExpand[Conjugate[A]]

For expressions within radicals, we must use the **Refine** command for the **ComplexExpand** command to work properly. Here is an example:

Without Refine command:

 In[]:= Simplify[ComplexExpand[Conjugate[Sqrt[a - b c]]]]
 Out[]:= ((a - b c)^2)^(1/4) (Cos[1/2 Arg[a - b c]] - I Sin[1/2 Arg[a - b c]])

With Refine command for $a-bc>0$ or $a-bc<0$

 In[]:= Refine[ComplexExpand[Conjugate[Sqrt[a - b c]]], 
 Assumptions -> a - b c > 0]
 Out[]:= Sqrt[a - b c]

 In[]:= Refine[ComplexExpand[Conjugate[Sqrt[a - b c]]], 
 Assumptions -> a - b c < 0]
 Out[]:= - I*Sqrt[b c - a]