To simplify the expression:

$\frac{x^4 + 3x^3 - 4x}{x^2 - x - 12}$

we can approach it step by step:

Step 1: Factor the denominator

We start by factoring the quadratic expression in the denominator:

$x^2 - x - 12$

We need two numbers that multiply to give $-12$ and add to give $-1$. These numbers are $-4$ and $3$. So, the denominator factors as:

$x^2 - x - 12 = (x - 4)(x + 3)$

Step 2: Simplify the numerator

The numerator is $x^4 + 3x^3 - 4x$. First, factor out $x$ from the entire expression:

$x(x^3 + 3x^2 - 4)$

Now, try to factor $x^3 + 3x^2 - 4$. Unfortunately, this doesn't factor easily, so we'll leave it as is.

Step 3: Write the expression in factored form

So the expression is now:

$\frac{x(x^3 + 3x^2 - 4)}{(x - 4)(x + 3)}$

At this point, there is no further simplification that can be done since no factors cancel out.

This is the simplified form of the expression. Would you like a deeper explanation or further steps?

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Relative Questions:

1. How do you factor higher-degree polynomials like $x^3 + 3x^2 - 4$?
2. What are some tips for recognizing when an expression can be factored?
3. Can polynomial long division help simplify complex expressions further?
4. What other methods exist for simplifying rational expressions?
5. When do factors in the numerator and denominator cancel out?

Tip: Always try to factor both the numerator and denominator first when simplifying rational expressions.