Question

If
$x = \sqrt[3]{2} + \sqrt[3]{4}$
Then Solve
${x}^{3} - 6x - 6 = 0$

Answer 1

Hope this helps. It is substitution.

Answer 2

Given, $\bold{x = \sqrt[3]{2}+\sqrt[3]{4}}$
x = ³√2 + ³√4 ------(1)
Taking cube both sides,
x³ = (³√2 + ³√4)³
x³ = (³√2)³ + (³√4)³ + 3(³√2)²(³√4) + 3(³√2)(³√4)² [∵ (a + b)³ = a³ + b³ + 3a²b + 3ab²]
x³ = 2 + 4 + 6³√2 + 6³√4 [∵(³√2)²(³√4) = (2)^(2/3) × (2)^(2/3) = (2)^4/3 = 2³√2 similarly ( ³√2)(³√4)² = 2³√4]
x³ = 6 + 6( ³√2 + ³√4)
x³ = 6 + 6x [ from equation (1) ]
x³ - 6x - 6 = 0