Certainly! Let's simplify and analyze the given expression step-by-step:
The question presents the rational function:
[tex]\[
\frac{7x^2 - 17x + 38}{(x + 6)(x - 1)^2}.
\][/tex]
### Step 1: Identify the numerator and the denominator
- The numerator of the given expression is:
[tex]\[
7x^2 - 17x + 38.
\][/tex]
- The denominator of the given expression is:
[tex]\[
(x + 6)(x - 1)^2.
\][/tex]
### Step 2: Factorize the numerator (if possible)
We need to consider if the numerator can be factorized. The given quadratic polynomial in the numerator is [tex]\(7x^2 - 17x + 38\)[/tex]. Factoring this kind of polynomial can be complex, and since we're treating the original given expression as simplified, we take it as it is. It does not appear to factor neatly into simpler polynomials with real coefficients.
### Step 3: Identify the structure of the denominator
The denominator is already given in factored form:
[tex]\[
(x + 6) \text{ and } (x - 1)^2.
\][/tex]
### Step 4: Simplify (if possible)
Given the constraints, simplifying the expression further might involve partial fraction decomposition, which breaks the expression into simpler parts. However, for this explanation, we consider the given structure as simplified and do not proceed to decompose further since no further factorization of the numerator is apparent.
### Step 5: Write down the simplified rational expression
Considering all the steps and recognizing that any further factorization didn't simplify the given expression beyond its current form, the final simplified form of the given rational expression is:
[tex]\[
\frac{7x^2 - 17x + 38}{(x + 6)(x - 1)^2}.
\][/tex]
This form gives a clear expression of the numerator and the factors of the denominator, making it suitable for further analysis or application in problems requiring this rational function.
From here, additional steps such as identifying vertical and horizontal asymptotes, intercepts, or domain restrictions can be determined based on this standard form.
