Given, cos^–1α + cos^–1β + cos^–1γ = 3π … (1)

Principal value of cos^-1 x is [0, π]

So, maximum value which cos^-1 x can have is π.

So, if (1) is correct then all the three terms i.e,

cos^–1α, cos^–1β, cos^–1γ should be equal to π

So, cos^–1α = π

cos^–1β = π

cos^–1γ = π

So, α = β = γ = -1

So, α(β + γ) + β (γ + α) + γ (α + β)

= (-1)(-1-1) + (-1)(-1-1) + (-1)(-1-1)

= 3(-1)(-2)

= 6