The expression \(\frac{8x^2 - 10x + 4}{x - 5}\) represents a rational function, where we aim to simplify or divide the polynomial \(8x^2 - 10x + 4\) by \(x - 5\). To do this, we need to perform polynomial long division or synthetic division. Let's proceed with polynomial long division.

1. Divide the first term of the numerator \(8x^2\) by \(x\), which gives \(8x\).
2. Multiply \(8x\) by \((x - 5)\), giving \(8x^2 - 40x\).
3. Subtract this result from the original polynomial: 
   \[
   (8x^2 - 10x + 4) - (8x^2 - 40x) = 30x + 4.
   \]
4. Now, divide \(30x\) by \(x\), which gives \(+30\).
5. Multiply \(30\) by \((x - 5)\), giving \(30x - 150\).
6. Subtract this result from the remainder:
   \[
   (30x + 4) - (30x - 150) = 154.
   \]

The final result is:
\[
\frac{8x^2 - 10x + 4}{x - 5} = 8x + 30 + \frac{154}{x - 5}.
\]

This is the simplified form of the given rational function.

Would you like to go over any specific steps in more detail?

Here are 5 related questions to deepen your understanding:
1. How does synthetic division differ from polynomial long division?
2. What is the remainder theorem, and how does it apply here?
3. How can we graph rational functions like this one?
4. What happens if we divide by a factor that doesn't exist in the numerator?
5. How do vertical asymptotes relate to the denominator of a rational function?

**Tip:** Always check for any common factors between the numerator and denominator before proceeding with division!

Learn how to simplify the rational function (8x^2 - 10x + 4)/(x - 5) using polynomial long division. This detailed step-by-step solution covers key algebraic concepts such as polynomial division and rational function simplification. Suitable for students in Grades 10-12.

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