Apologies for the confusion. Let's go through the simplification process again.
We have the expression: x^3 - 2x^2 + 1 + (3x^5 - 12x^4 + 21x^2) / (x^2 - 4x).
First, let's simplify the fraction:
(3x^5 - 12x^4 + 21x^2) / (x^2 - 4x)
We can factor out an x^2 from the numerator:
x^2(3x^3 - 12x^2 + 21) / (x^2 - 4x)
Next, we can factor out an x from the numerator:
x^2(x(3x^2 - 12x + 21)) / (x(x - 4))
Now, we can cancel out the common factors of x in the numerator and denominator:
(x(3x^2 - 12x + 21)) / (x - 4)
Thus, the simplified expression is:
x^3 - 2x^2 + 1 + (x(3x^2 - 12x + 21)) / (x - 4)
Next, let's simplify the expression further:
x^3 - 2x^2 + 1 + (x(3x^2 - 12x + 21)) / (x - 4)
To add the fractions, we need a common denominator. Multiplying (x - 4) both numerator and denominator of the second fraction gives:
x^3 - 2x^2 + 1 + [x(3x^2 - 12x + 21)(x - 4)] / (x(x - 4))
Expanding the numerator of the second fraction, the expression becomes:
x^3 - 2x^2 + 1 + [x(3x^3 - 12x^2 + 21x - 12x^2 + 48x - 84)] / (x(x - 4))
Simplifying further:
x^3 - 2x^2 + 1 + [x(3x^3 - 24x^2 + 69x - 84)] / (x(x - 4))
Combining like terms in the second fraction of the numerator:
x^3 - 2x^2 + 1 + [3x^4 - 24x^3 + 69x^2 - 84x] / (x(x - 4))
Now, let's combine like terms in the numerator:
x^3 - 2x^2 + 1 + 3x^4 - 24x^3 + 69x^2 - 84x / (x(x - 4))
Rearranging the terms:
3x^4 - 24x^3 + x^3 + 69x^2 - 2x^2 - 84x + 1 / (x(x - 4))
Combining like terms in the numerator:
3x^4 - 23x^3 + 67x^2 - 84x + 1 / (x(x - 4))
Thus, the completely simplified expression is:
3x^4 - 23x^3 + 67x^2 - 84x + 1 / (x(x - 4))