Question

what is the difference?
\\(\frac{x + 5}{x + 2} - \frac{x + 1}{x^2 + 2x}\\)
\\(\bigcirc\\) \\(\frac{x^2 + 4x - 1}{x(x + 2)}\\)
\\(\bigcirc\\) \\(\frac{x^2 + 4x + 1}{x(x + 2)}\\)
\\(\bigcirc\\) \\(\frac{4}{-1(x^2 + x - 2)}\\)
\\(\bigcirc\\) \\(\frac{x^2 + 6x + 1}{x(x + 2)}\\)

Explanation:

Step1: Factor the denominator of the second fraction

The second fraction's denominator is \(x^2 + 2x\), which factors to \(x(x + 2)\). So the expression becomes \(\frac{x + 5}{x + 2}-\frac{x + 1}{x(x + 2)}\).

Step2: Find a common denominator

The common denominator for the two fractions is \(x(x + 2)\). Multiply the first fraction by \(\frac{x}{x}\) to get a common denominator: \(\frac{(x + 5)x}{x(x + 2)}-\frac{x + 1}{x(x + 2)}\).

Step3: Expand the numerator of the first fraction

Expand \((x + 5)x\) to get \(x^2+5x\). Now the expression is \(\frac{x^2 + 5x}{x(x + 2)}-\frac{x + 1}{x(x + 2)}\).

Step4: Subtract the numerators

Subtract the second numerator from the first: \(\frac{x^2 + 5x-(x + 1)}{x(x + 2)}\).

Step5: Simplify the numerator

Simplify \(x^2 + 5x-(x + 1)\) by distributing the negative sign: \(x^2+5x - x - 1=x^2 + 4x - 1\). So the result is \(\frac{x^2 + 4x - 1}{x(x + 2)}\).

Answer:

\(\frac{x^2 + 4x - 1}{x(x + 2)}\) (the first option: \(\boldsymbol{\frac{x^2 + 4x - 1}{x(x + 2)}}\))