x2+x−2=(x–1)(x+2) p(x)=x4+3x3+ax2+bx+16 Since (x2+x−2) is a factor of p(x), and (x−1) and (x+2) are factors of (x2+x−2), then p(x) is divisible by (x–1) and (x+2).
p(1)=1+3+a+b+16=0⇒a+b=−20....(i)
p(−2)=16−24+4a−2b+16=0 ⇒4a−2b=−8⇒2a−b=−4...(ii) (i) + (ii) gives 3a=–24⇒a=–8 Substituting it in (i) we get b=−12 ∴(a,b)=(−8,−12)