What is obtained by simplifying \(\frac{x^4+2x}{x}\) for \(x\neq0\)?

Correct answer: \(x^3+2\)

The direct answer is A: \\(x^3+2\\). Since \\(x\\neq0\\), division by x is permitted. Split the numerator term by term: \\(\\frac{x^4+2x}{x}=\\frac{x^4}{x}+\\frac{2x}{x}\\). Using \\(x^m/x=x^{m-1}\\) for nonzero x, \\(x^4/x=x^3\\). Also, \\(2x/x=2\\). Therefore the result is \\(x^3+2\\). Option A is correct. Option B, \\(x^4+2\\), fails to reduce the first power and incorrectly leaves the second term as 2 after removing x inconsistently. Option C, \\(x^3+2x\\), simplifies the first term but fails to cancel x in the second term. Option D, \\(x^2+2\\), subtracts 2 from the exponent instead of 1 in the first term, so it is incorrect. The condition \\(x\\neq0\\) matters because division by zero is not defined. One may also factor the numerator: \\(x^4+2x=x(x^3+2)\\), and then cancel x to get \\(x^3+2\\). Memory cue: divide each term by x, reducing the exponent by one when the term contains x.