1. **State the problem:** Simplify or factor the expression $\frac{4x^2 + x}{4x^2 - 7x - 2}$. 2. **Write the expression:** $$\frac{4x^2 + x}{4x^2 - 7x - 2}$$ 3. **Factor numerator and denominator:** - Numerator: $4x^2 + x = x(4x + 1)$ - Denominator: Factor $4x^2 - 7x - 2$. Find two numbers that multiply to $4 \times (-2) = -8$ and add to $-7$. These are $-8$ and $1$. Rewrite denominator: $$4x^2 - 8x + x - 2 = (4x^2 - 8x) + (x - 2) = 4x(x - 2) + 1(x - 2) = (4x + 1)(x - 2)$$ 4. **Rewrite the fraction:** $$\frac{x(4x + 1)}{(4x + 1)(x - 2)}$$ 5. **Cancel common factors:** $$\frac{x\cancel{(4x + 1)}}{\cancel{(4x + 1)}(x - 2)} = \frac{x}{x - 2}$$ 6. **Final answer:** $$\boxed{\frac{x}{x - 2}}$$ This is the simplified form of the original expression, valid for $x \neq \frac{-1}{4}$ and $x \neq 2$ (to avoid division by zero).