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Solve:       ( x² + x + 1 ) ( x² + x + 2 )  =  12

Multiply out:  x⁴ + 2x³ + 4x² + 3x + 2  =  12

Rewrite:       x⁴ + 2x³ + 4x² + 3x - 10  =  0

 

There are two rational solutions; you can find them by trying all the factors of -10:

     1, -1, 2, -2, 5, -5, 10, -10          (two of these will work)

 

After you have these two solutions, you can divide their factors out of the original problem.

This will give you a quadratic that can be solved by using the quadratic equation.