Sure, let's tackle the problem of simplifying the rational expression:
[tex]\[
\frac{x^3 - 4x^2 + 5x - 20}{x - 4}
\][/tex]
Here are the steps to simplify it:
1. Recognize the form:
We have a polynomial in the numerator [tex]\(x^3 - 4x^2 + 5x - 20\)[/tex] and a linear term in the denominator [tex]\(x - 4\)[/tex].
2. Polynomial Long Division:
Use polynomial long division to divide the numerator by the denominator. However, for simplicity, we use factorization and cancel common factors.
3. Factor the Numerator:
Let's assume we already factored the polynomial [tex]\(x^3 - 4x^2 + 5x - 20\)[/tex]. After factorization, it simplifies to:
[tex]\[
(x - 4)(x^2 + 5)
\][/tex]
4. Cancel Common Factors:
The expression now becomes:
[tex]\[
\frac{(x - 4)(x^2 + 5)}{x - 4}
\][/tex]
Since [tex]\(x \neq 4\)[/tex], we can cancel the common factor [tex]\((x - 4)\)[/tex] from the numerator and the denominator:
[tex]\[
x^2 + 5
\][/tex]
5. Simplified Expression:
Therefore, the simplified form of the given rational function is:
[tex]\[
x^2 + 5
\][/tex]
After these steps, we conclude that the correct option is:
A. [tex]\(x^2 + 5\)[/tex]
So the final simplified expression is:
[tex]\[
\boxed{x^2 + 5}
\][/tex]
