The expression you've given is a rational expression:

\[
\frac{15x^2 + 14x - 8}{3x + 4}
\]

To simplify this expression, let's see if we can factor the numerator and denominator and then cancel out any common factors.

### Step 1: Factor the Numerator
First, we factor the quadratic expression \(15x^2 + 14x - 8\). We look for two numbers that multiply to \(15 \times (-8) = -120\) and add up to \(14\).

The factors of \(-120\) that add up to \(14\) are \(20\) and \(-6\).

So, we can rewrite the middle term as follows:

\[
15x^2 + 20x - 6x - 8
\]

Now, we can factor by grouping:

\[
(15x^2 + 20x) - (6x + 8)
\]

Factor out the common factors in each group:

\[
5x(3x + 4) - 2(3x + 4)
\]

Now, factor out the common binomial factor \((3x + 4)\):

\[
(5x - 2)(3x + 4)
\]

So, the numerator \(15x^2 + 14x - 8\) factors as \((5x - 2)(3x + 4)\).

### Step 2: Simplify the Expression
Now that we have the factorized form, substitute it back into the original expression:

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

We see that the \((3x + 4)\) terms in the numerator and denominator cancel out, leaving:

\[
5x - 2
\]

### Final Answer
So, the simplified form of the given expression is:

\[
5x - 2
\]

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you factor a quadratic expression like \(ax^2 + bx + c\)?
2. What are some common techniques for simplifying rational expressions?
3. What happens if the factor in the denominator is a repeated factor in the numerator?
4. Can you always cancel terms in a rational expression?
5. How do you solve a quadratic equation by factoring?

**Tip:** Always check for common factors in both the numerator and the denominator before simplifying a rational expression.

This page provides a detailed solution for simplifying the rational expression 15x^2 + 14x - 8 / 3x + 4 by factoring and canceling common factors. It covers essential algebraic concepts such as factoring quadratic expressions and simplifying rational expressions. Suitable for students in Grades 9-10.

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