The expression provided is:

$\frac{15x^2 + x - 1}{5x - 3}$

We are tasked with simplifying or dividing this rational expression.

Step-by-step solution:

1. Factor the numerator: The numerator is $15x^2 + x - 1$. We can attempt to factor this quadratic expression using the method of splitting the middle term or other techniques.

   We look for two numbers that multiply to $15 \times -1 = -15$ and add to $1$. These numbers are $3$ and $-5$.

   So, we rewrite $x$ as: $15x^2 + 3x - 5x - 1$

2. Group terms and factor: Group the terms as: $(15x^2 + 3x) - (5x + 1)$ Factor each group: $3x(5x + 1) - 1(5x + 1)$

3. Factor the common binomial: The common binomial is $(5x + 1)$, so we factor it out: $(5x + 1)(3x - 1)$

4. Simplify the rational expression: Now, the expression becomes: $\frac{(5x + 1)(3x - 1)}{5x - 3}$

There are no common factors between the numerator and the denominator, so the expression cannot be simplified further.

Thus, the final simplified expression is: $\frac{(5x + 1)(3x - 1)}{5x - 3}$

Would you like more details on any step?

Relative Questions:

1. How do you factor quadratics more effectively?
2. What is the method of splitting the middle term in factoring?
3. How do you simplify rational expressions when both numerator and denominator are factorable?
4. Can you always factor quadratics? What if they are irreducible?
5. How can you apply factoring techniques to solve quadratic equations?

Tip:

When factoring a quadratic expression, always check if the discriminant ($b^2 - 4ac$) is a perfect square to ensure it can be factored over the real numbers.