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Given the function:

\[
k(x) = \frac{x^3 + 4x^2 + x - 5}{x^2 + 5x + 4}
\]

### Step 1: Simplify the Function

To simplify, let's see if the numerator \(x^3 + 4x^2 + x - 5\) can be factored or simplified with the denominator \(x^2 + 5x + 4\).

**Factor the denominator:**

1. \( x^2 + 5x + 4 = (x + 1)(x + 4) \).

**Factor the numerator:**

The numerator, \(x^3 + 4x^2 + x - 5\), can be simplified by factoring or using synthetic division.

Let's try dividing the numerator by one of the factors of the denominator.

### Step 2: Synthetic Division

Let's perform synthetic division of the numerator \(x^3 + 4x^2 + x - 5\) by one of the factors, \(x + 1\):

1. Coefficients of the numerator are: \(1, 4, 1, -5\).
2. Divide by \(x + 1\), meaning we use \(-1\).

Performing the division:

\[
\begin{array}{r|rrrr}
-1 & 1 & 4 & 1 & -5 \\
   &   & -1 & -3 & 2 \\
\hline
   & 1 & 3 & -2 & -3 \\
\end{array}
\]

The quotient is \(x^2 + 3x - 2\) with a remainder of \(-3\).

So, \(k(x)\) can be expressed as:

\[
k(x) = (x^2 + 3x - 2) + \frac{-3}{x^2 + 5x + 4}.
\]

### Step 3: Find the Domain

The domain excludes values that make the denominator zero:

Set \(x^2 + 5x + 4 = 0\).

Solving this gives:

\[
x = -1 \quad \text{and} \quad x = -4.
\]

So, the domain of \(k(x)\) is:

\[
x \in \mathbb{R}, \quad x \neq -1, \quad x \neq -4.
\]

Would you like further details or have any questions?

### Related Questions:
1. How to perform synthetic division in general?
2. What are the steps to factor polynomials efficiently?
3. How to determine the vertical and horizontal asymptotes of rational functions?
4. How to find and interpret the domain of complex functions?
5. How can you use factoring to simplify complex fractions?

**Tip:** When simplifying rational functions, always check for factorable parts in the numerator and denominator to simplify expressions effectively!

k(x) = (x^3 + 4x^2 + x - 5) / (x^2 + 5x + 4)

Learn how to simplify the rational function k(x) = (x^3 + 4x^2 + x - 5) / (x^2 + 5x + 4) using synthetic division and factoring. This guide explains key algebraic concepts like polynomial division, factoring, and finding the domain. Suitable for high school students in Grades 10-12.

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