Question 38041
The answer key is wrong.  It happens every now and then...professors write these books and depend on overworked and underpaid graduate assistants to edit them...
You're okay so far...
3x^6 - 192 =
3(x^6 - 64) =
3(x^3 - 8)(x^3 + 8)
Now from here, there is a not-too-well-known way to factor the sum and difference of cubes...you'll have to memorize the method...but here it is...
a^3 + b^3 = (a + b)(a^2 - ab + b^2) and
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
so in our case
3(x^3 - 8)(x^3 + 8) =
3(x - 2)(x^2 + 2x + 4)(x + 2)(x^2 - 2x + 4)
and we're done...
And by the way, you should be embarassed if you stop wishing to learn, but proud of yourself for forging on ahead!