You have shared the following rational expression:

$\frac{9x^2 + 20x - 10}{(x + 2)(3x - 1)}$

To simplify this, we need to check if the quadratic in the numerator can be factored in such a way that it shares a common factor with the denominator.

Step 1: Factor the numerator $9x^2 + 20x - 10$

We are looking for two numbers that multiply to $9 \times (-10) = -90$ and add to $20$.

• The factors of $-90$ that add up to $20$ are $30$ and $-3$. Thus, we rewrite $9x^2 + 20x - 10$ as: $9x^2 + 30x - 3x - 10$

Step 2: Factor by grouping

Group the terms as follows: $(9x^2 + 30x) - (3x + 10)$ Now factor each group: $3x(3x + 10) - 1(3x + 10)$

We can factor out $(3x + 10)$: $(3x + 10)(3x - 1)$

Step 3: Simplify the expression

Now the expression becomes: $\frac{(3x + 10)(3x - 1)}{(x + 2)(3x - 1)}$

Since $(3x - 1)$ appears in both the numerator and the denominator, we can cancel it out: $\frac{3x + 10}{x + 2}$

Thus, the simplified form of the expression is: $\frac{3x + 10}{x + 2}$

Let me know if you'd like further details or have any questions!

Here are 5 related questions to expand on this:

1. What happens if the quadratic expression in the numerator cannot be factored?
2. How would you solve the equation if the entire rational expression were set equal to zero?
3. What are the restrictions on the variable $x$ for the original expression?
4. Can this simplification help with graphing this rational function? If so, how?
5. How would you determine the vertical and horizontal asymptotes for this rational function?

Tip: Always check for restrictions in the denominator of a rational function, as these indicate values where the function is undefined.