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To find the horizontal and vertical asymptotes for the given rational function:

\[
y = \frac{2x^2 + x - 1}{x^2 + x - 90}
\]

### 1. **Horizontal Asymptote**
Horizontal asymptotes depend on the degrees of the polynomials in the numerator and the denominator.

- The degree of the numerator \(2x^2 + x - 1\) is 2.
- The degree of the denominator \(x^2 + x - 90\) is also 2.

When the degrees of the numerator and denominator are the same, the horizontal asymptote is found by taking the ratio of the leading coefficients.

- Leading coefficient of the numerator: \(2\)
- Leading coefficient of the denominator: \(1\)

Thus, the horizontal asymptote is:

\[
y = \frac{2}{1} = 2
\]

### 2. **Vertical Asymptote**
Vertical asymptotes occur where the denominator equals zero (and the numerator does not simultaneously equal zero at the same value).

Set the denominator equal to zero and solve for \(x\):

\[
x^2 + x - 90 = 0
\]

This quadratic equation can be solved using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 1\), \(b = 1\), and \(c = -90\). Substituting these values in:

\[
x = \frac{-1 \pm \sqrt{1 + 360}}{2} = \frac{-1 \pm \sqrt{361}}{2} = \frac{-1 \pm 19}{2}
\]

So, the solutions are:

\[
x = \frac{18}{2} = 9 \quad \text{and} \quad x = \frac{-20}{2} = -10
\]

Therefore, the vertical asymptotes are:

\[
x = 9 \quad \text{and} \quad x = -10
\]

### Summary
- **Horizontal Asymptote:** \(y = 2\)
- **Vertical Asymptotes:** \(x = 9\) and \(x = -10\)

Would you like more details or have any questions?

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**Related Questions:**
1. How do you find oblique asymptotes?
2. What happens if the degree of the numerator is higher than the denominator?
3. Can there be a situation where a function has no asymptotes at all?
4. How can horizontal asymptotes be used to understand the end behavior of a function?
5. What is the significance of the roots of the numerator in determining the asymptotes?

**Tip:** Always check the numerator when identifying vertical asymptotes, as common factors between numerator and denominator can remove potential asymptotes.

Find the horizontal and vertical asymptotes of the curve: y = (2x^2 + x - 1) / (x^2 + x - 90).

Learn how to find the horizontal and vertical asymptotes of the rational function y = (2x^2 + x - 1) / (x^2 + x - 90). This problem involves key algebraic concepts like rational functions, asymptotes, and the quadratic formula. Suitable for Grades 10-12 students.

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