Certainly! Let's simplify the given expression step-by-step:
The given expression is:
[tex]\[
\frac{x^4 - x^2 + 9}{x^3 - x^2}
\][/tex]
Step 1: Factor the denominator.
The denominator [tex]\(x^3 - x^2\)[/tex] can be factored by taking out the common factor [tex]\(x^2\)[/tex]:
[tex]\[
x^3 - x^2 = x^2(x - 1)
\][/tex]
Step 2: Substitute the factorized form of the denominator back into the original expression:
[tex]\[
\frac{x^4 - x^2 + 9}{x^2(x - 1)}
\][/tex]
Step 3: Examine the numerator [tex]\(x^4 - x^2 + 9\)[/tex].
The numerator, in this case, does not appear to factor in a straightforward manner using simple algebraic manipulation.
Step 4: Since the numerator does not factor in a simple way, we leave it as it is and recognize that the expression cannot be simplified further in terms of factoring.
So the simplified form of the given expression is:
[tex]\[
\frac{x^4 - x^2 + 9}{x^2(x - 1)}
\][/tex]
Thus, the final simplified expression is:
[tex]\[
\frac{x^4 - x^2 + 9}{x^2(x - 1)}
\][/tex]
