Question 1104188
I found it by trial and error, using hints from  {{{ f(g(x)) = 4x^2 + 10x + 5 }}}

I assumed  {{{ g(x) = ax + b }}}    
where a=2 because we need to end up with {{{4x^2}}}
and   2ab - a = 10  —>  a(2b-1) = 10  (EDIT: because a=2, this implies b=3, missed that earlier)
Another hint/constraint is that  {{{ b^2 - b - 1 = 5 }}} 

I found a=2 and b=3 works.   {{{ g(x) = 2x + 3 }}} 
                                                     
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Of course, this worked out because I guessed the proper form for g(x).  
Maybe another tutor may have a more structured method.   I'm often amazed at the talent base here.
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