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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]