What will be value of f(x) if \(\begin{vmatrix}2ab & a^2 & b^2 \\a^2 & b^2 & 2ab \\b^2 & 2ab & a^2 \end {vmatrix}\)?

Given,\(\begin{vmatrix}2ab & a^2 & b^2 \\a^2 & b^2 & 2ab \\b^2 & 2ab & a^2 \end {vmatrix}\)
Using C₁ = C₁ + C₂ + C₃
= \(\begin{vmatrix}a^2 + b^2 + 2ab & a^2 & b^2 \\a^2 + b^2 + 2ab & b^2 & 2ab \\a^2 + b^2 + 2ab & 2ab & a^2 \end {vmatrix}\)
= (a + b)²\(\begin{vmatrix}1 & a^2 & b^2 \\1 & b^2 & 2ab \\1 & 2ab & a^2 \end {vmatrix}\)
= (a + b)²\(\begin{vmatrix}1 & a^2 & b^2 \\1 & b^2 – a^2 & 2ab – b^2 \\0 & 2ab – a^2 & a^2 – b^2 \end {vmatrix}\)
= (a + b)²[(b² – a²)(a² – b²) ? (2ab – b²)( 2ab – a²)]
= -(a + b)²[(a² – b²)² + 4a²b² ? 2ab(a² + b²) + a² b²)]
= ?(a + b)²[(a²+b²)² ? 2(a²+b²) (ab)+(ab)²]
= ?(a + b)²(a² + b² – ab)²
= ?[(a + b)²(a² + b² – ab)²]²
= ?(a³ + b³)²