C1 → C1 + C2 + C3 (i.e. replace 1^st column by addition of 1^st, 2^nd and 3^rd column)

Taking 2(x + y + z) outside the determinant from 1^st column

R2 → R2 - R1 (i.e. replace 2^nd row by subtraction of 2^nd and 1^st row)

R3 → R3 - R1 (i.e. replace 3^rd row by subtraction of 3^rd and 1^st row)

Expanding the determinant along 1^st column

∴ LHS = 2(x + y + z)³ = RHS

∴