Let ( u = x^3 + 1 ).

Then, the equation becomes:

( u^2 - 10u + 9 = 0 )

This is a quadratic equation that can be factored as:

( (u-9)(u-1) = 0 )

Setting each factor to zero:

( u - 9 = 0 ) or ( u - 1 = 0 )

( u = 9 ) or ( u = 1 )

Now substitute back in ( x^3 + 1 ) for ( u ):

For ( u = 9 ):

( x^3 + 1 = 9 )

( x^3 = 8 )

( x = 2 )

For ( u = 1 ):

( x^3 + 1 = 1 )

( x^3 = 0 )

( x = 0 )

Therefore, the solutions to the original equation are ( x = 2 ) and ( x = 0 ).