Prove that 
is divisible by a + b + c and find the quotient.
We have, 
Applying R1→ R1 – R2, we get
Taking (a + b+ c) common from first row, we get
Applying R2→ R2 – R3, we get
Taking (a + b+ c) common from second row, we get
Applying C1→ C1 + C2 + C3, we get
Now, expanding along C1, we get
= (a + b + c)2[ab + bc + ca – (a2 + b2 + c2){(c – b)(b – a) – (a – c)2}]
= (a + b + c)2[ab + bc + ca – (a2 + b2 + c2){(cb – ac – b2 + ab – (a + c2 – 2ac)}]
= (a + b + c)2[ab + bc + ca – (a2 + b2 + c2){(cb – ac – b2 + ab – a - c2 + 2ac)}]
= (a + b + c)2[ab + bc + ca – (a2 + b2 + c2){ac + bc + ab – (a2 + b2 + c2)}]
=(a + b + c)2[ab + bc + ca – (a2 + b2 + c2)]2
=(a + b + c)(a + b + c)[ab + bc + ca – (a2 + b2 + c2)
Hence, given determinant is divisible by (a + b + c) and Quotient is (a + b + c)[ab + bc + ca – (a2 + b2 + c2)