Rationale
2x/(4x² + x) simplifies to 2/(4x + 1).
By factoring the denominator, we find that 4x² + x can be rewritten as x(4x + 1), allowing us to simplify the expression to 2/(4x + 1). This shows that the correct choice is indeed C, represented in a simplified form.
A) The expression 1/x + 2 does not match the simplified form of the given rational expression. This incorrect choice suggests a different relationship that does not arise from simplifying the original equation, as it introduces a new variable and structure that do not correspond to the simplification process applied to 2x/(4x² + x).
B) The choice 1/2x + 1 implies a completely different operation, suggesting a sum of fractions rather than a simplification of the given rational expression. This form does not correctly reflect the original structure or provide an equivalent expression when compared to 2x/(4x² + x).
C) The expression 2/(4x + 1) accurately represents the simplified form of 2x/(4x² + x) after factoring out x from the denominator. This matches the derived expression and confirms that it is the correct choice.
D) The option 2/4x² + 1 suggests a numerator that is not equivalent to the original expression's simplification. This format incorrectly implies a different relationship that results from a misunderstanding of the initial rational expression and does not hold as a valid equivalence.
Conclusion
The simplification of the expression 2x/(4x² + x) leads us directly to the result 2/(4x + 1), confirming that option C is the correct choice. The other options either introduce incorrect relationships or fail to represent the original expression accurately, emphasizing the importance of proper factorization and simplification techniques in algebra.