Correct Answer: –6

Solution :

(a + b + c) ² = a ² + b ² + c ² + 2ab + 2bc + 2ca
⇒ 6 ² = 38 + 2ab + 2bc + 2ca
⇒ (ab + bc + ca) = $\frac{(36 – 38)}{2}$
⇒ (ab + bc + ca) = – 1
Now, a(b ² + c ² ) + b(c ² + a ² ) + c(a ² + b ² ) + 3abc
= a(38 – a ² ) + b(38 – b ² ) + c(38 – c ² ) + 3abc
= 38(a + b + c) – {(a ³ + b ³ + c ³ ) – 3abc}
= (38 × 6) – {(a + b + c)(a ² + b ² + c ² – ab – bc – ca)}
= (38 × 6) – {6 × (38 – (–1))}
= (38 × 6) – 6 × 39
= 6(38 – 39)
= – 6
Hence, the correct answer is –6.