Hi, I was confused if I was factoring high number exponents completely

For example: x^{8}-22x^{7}+121x^{6}

I just made it into a simple polinomial inside, by doing x^{6}(x^{2}-22x+121)

Then I factored it down to xIs this really factoring it down completely?^{6}(x-11)(x-11) which also equals x^{6}(x-11)^{2 }

Because I did it with other polynomials such as x^{3}+16x^{2}+16x and turned it intoxAs well with^{2}(x+8)(x+8)

4x^{6}-12x^{5}+9x^{4}and factored it down to x^{4}(2x-3)(2x-3)

Am I really factoring it down completely? Because I did this with all my problems, taking out the x with the lowest exponents then turning the ones into the paranthesis into simple polynomials and factoring that instead.

Thanks