2016/046) Show that there is no combination of integers a,b,c for which $f(n) = n^3 + an^2 + bn+ c$ is perfect square for all n

If we can show that it cannot be for n = 1,2,3,4 then we are through
for n = 1 we have $f(1) = 1 + a + b+ c\cdots(1)$
for n= 2 we have $f(2) = 8 + 4a + 2b + c\cdots(2)$
$f(3) = 27 + 9a + 3b + +c\cdots(3)$
$f(4) = 64 + 16a + 4b + c\cdots(4)$
if f(1) and f(3) are perfect squares then from (1) and (3) we get $26+ 8a+ 2b \equiv 0 \pmod 4$ or $2b \equiv 2 \pmod 4\cdots(5)$
if f(2) and f(4) are perfect squares then from (2) and (4) we get $56+ 12a+ 2b \equiv 0 \pmod 4$ or $2b \equiv 0 \pmod 4\cdots(6)$
(5) and (6) cannot be satisfied at the same time so for n = 1 to 4 it cannot be a perfect square