LHs=∣∣
∣
∣
∣
∣
∣
∣∣(a+b)2ccca(b+c)2aabb(c+a)2b∣∣
∣
∣
∣
∣
∣
∣∣=1abc∣∣
∣
∣∣(a+b)2c2c2a2(b+c)2a2b2b2(c+a)2∣∣
∣
∣∣
On applying C2→C2−C1andC3→C3−C1, we get-
=1abc∣∣
∣
∣∣(a+b)2c2−(a+b)2c2−(a+b)2a2(b+c)2−a20b20(c+a)2−b2∣∣
∣
∣∣
=1abc∣∣
∣
∣∣(a+b)2(a+b+c)(c−a−b)(a+b+c)(c−a−b)a2(a+b+c)(b+c−a)0b20(a+b+c)(c+a−b)∣∣
∣
∣∣
Taking (a+b+c) common from C2andC3,we get-
(a+b+c)2abc∣∣
∣
∣∣(a+b)2c−a−bc−a−ba2b+c−a0b20c+a−b∣∣
∣
∣∣
=(a+b+c)2abc∣∣
∣
∣∣(a+b)2c−a−bc−a−ba2b+c−a0b20c+a−b∣∣
∣
∣∣
On applying R1→R1−(R2+R3), we get-
=(a+b+c)2abc∣∣
∣
∣∣2ab−2b−2aa2b+c−a0b20c+a−b∣∣
∣
∣∣
On applying C3→C3+1bC1, we get-
(a+b+c)2abc∣∣
∣
∣
∣∣2ab−2b0a2b+c−aa2bb20c+a∣∣
∣
∣
∣∣
Expanding the determinant now-
(a+b+c)2abc[2ab(bc+ab+c2+ac−ac−a2)+2b(a2c+a3−a2b)]
On simplifying, we get-
(a+b+c)2abc×2abc(a+b+c)
=2(a+b+c)3
Hence,proved.