Question: User: $x - 3$
$\frac{2x}{x^2 + 7x - 4}$
Graph shape: a red marker/pen is pointing at the top of the fraction bar over the expression; position_hint: top-center.
1. **State the problem:** Simplify or analyze the expression $$\frac{2x}{x^2 + 7x - 4}$$ where the numerator is $2x$ and the denominator is $x^2 + 7x - 4$.
2. **Factor the denominator if possible:** We look for factors of $x^2 + 7x - 4$. Since the quadratic does not factor nicely with integers, we use the quadratic formula:
$$x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{-7 \pm \sqrt{49 + 16}}{2} = \frac{-7 \pm \sqrt{65}}{2}$$
3. **Domain restrictions:** The denominator cannot be zero, so $x \neq \frac{-7 + \sqrt{65}}{2}$ and $x \neq \frac{-7 - \sqrt{65}}{2}$.
4. **Simplify the expression:** The numerator is $2x$. There are no common factors with the denominator since it does not factor nicely, so the expression is already in simplest form.
5. **Summary:** The expression $$\frac{2x}{x^2 + 7x - 4}$$ is simplified as is, with domain restrictions excluding the roots of the denominator.
**Final answer:** $$\frac{2x}{x^2 + 7x - 4}$$ with $x \neq \frac{-7 \pm \sqrt{65}}{2}$.