R1 → R1 + R2 + R3 (i.e. replace 1^st row by addition of 1^st, 2^nd and 3^rd row)

Taking a + b + c outside the determinant from 1^st row

C2 → C2 - C1 (i.e. replace 2^nd column by subtraction of 2^nd and 1^st column)

C3 → C3 - C1 (i.e. replace 3^rd column by subtraction of 3^rd and 1^st column)

Expanding the determinant along 1^st row

∴ LHS = (a + b + c)³ = RHS

∴