Yes, Prof. Blinder is perfectly right! The remaining problem is to make a transcription of your picture of code, and then:
sysEq = {{a0 + a1 + a2 + a3 + a4}, {a1 + 3 a2 + 5 a3 + 7 a4}, {a1 +
9 a2 + 25 a3 + 49 a4 + 8 c0}, {a1 + 27 a2 + 125 a3 +
343 a4}} == {0, 0, 0, 0};
sysSol = Solve[sysEq, {a1, a2, a3, a4}];
coeff = First[{a0, a1, a2, a3, a4, c0} /. sysSol]
Cheers Henrik