P[u_, a_, b_, c_, d_]  :=  a*u^3 + b*u^2 +  c*u  +  d

v  =  
      Solve[
{P[0, a, b,  c, d]  ==  
            P0, 
P[1/4, a, b,  c, d] == 
            P1, 
 
          P[3/4, a, b, c, d] == 
            P3, 
P[1, a, b, c, d] == 
            P4}, 
{a, b,  c, d}]

v

v[[1]]

P[1/2, a, b, c, d]  /.   v[[1]]

Simplify[%]