Question

which expression is equal to \\(\frac{3x}{x + 3}+\frac{x}{x + 2}\\)? \\(\bigcirc\\) \\(\frac{3x^{2}+10x}{(x + 3)(x + 2)}\\) \\(\bigcirc\\) \\(\frac{4x^{2}+9x}{(x + 3)(x + 2)}\\) \\(\bigcirc\\) \\(\frac{4x^{2}+3x}{(x + 3)(x + 2)}\\) \\(\bigcirc\\) \\(\frac{x^{2}+12x}{(x + 3)(x + 2)}\\)

Explanation:

Step1: Find common denominator

The common denominator is $(x+3)(x+2)$. Rewrite each fraction:
$\frac{3x}{x+3} = \frac{3x(x+2)}{(x+3)(x+2)}$, $\frac{x}{x+2} = \frac{x(x+3)}{(x+3)(x+2)}$

Step2: Expand numerators

Expand the terms in the numerators:
$3x(x+2) = 3x^2 + 6x$, $x(x+3) = x^2 + 3x$

Step3: Add the numerators

Combine the expanded numerators over the common denominator:
$\frac{(3x^2 + 6x) + (x^2 + 3x)}{(x+3)(x+2)}$

Step4: Simplify the numerator

Combine like terms in the numerator:
$3x^2 + x^2 + 6x + 3x = 4x^2 + 9x$

Answer:

$\frac{4x^2+9x}{(x+3)(x+2)}$ (the second option)