k
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=
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4
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5
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4

The given function is:

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

To analyze this function, let’s first examine its structure:

1. **Numerator**: \(x^3 + 4x^2 + x - 5\)
2. **Denominator**: \(x^2 + 5x + 4\)

### Factoring the Denominator:
We will try to factor the denominator \(x^2 + 5x + 4\):
\[
x^2 + 5x + 4 = (x + 1)(x + 4)
\]

### Attempt to Factor the Numerator:
Now, we'll check if the numerator can be factored. We can try factoring or use synthetic division if it’s not easily factorable.

The goal will be to see if the function can be simplified. Would you like me to proceed with factoring the numerator or synthetic division to simplify the function further?

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Here are some related questions to explore:
1. How do you factor a cubic polynomial?
2. How do you perform synthetic division on a polynomial?
3. What are vertical asymptotes, and how can we find them in rational functions?
4. How do you find the roots of a cubic function?
5. How can you check for any holes or removable discontinuities in the function?

**Tip**: Always check if a polynomial can be factored before applying synthetic division or other simplification methods.

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

Explore how to simplify the rational function k(x) = (x^3 + 4x^2 + x - 5) / (x^2 + 5x + 4) by factoring and using synthetic division. This solution covers key algebraic concepts, such as factoring polynomials and rational function simplification. Suitable for high school students in Grades 10-12.

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k ( x )

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