Find the sum of the terms. \( \frac{x}{x^{2}+3 x+2}+\frac{3}{x+1} \) The numerator of the simplified sum is

To simplify the expression \( \frac{x}{x^{2}+3x+2}+\frac{3}{x+1} \), we first factor the denominator of the first term. The quadratic \( x^{2}+3x+2 \) factors to \( (x+1)(x+2) \). So the first term can be rewritten as: \[ \frac{x}{(x+1)(x+2)} + \frac{3}{x+1} \] Next, we need a common denominator to combine these fractions. The common denominator is \( (x+1)(x+2) \). We rewrite the second term accordingly: \[ \frac{3}{x+1} = \frac{3(x+2)}{(x+1)(x+2)} = \frac{3x + 6}{(x+1)(x+2)} \] Now, we can add the two terms: \[ \frac{x}{(x+1)(x+2)} + \frac{3x + 6}{(x+1)(x+2)} = \frac{x + 3x + 6}{(x+1)(x+2)} = \frac{4x + 6}{(x+1)(x+2)} \] The numerator of the simplified sum is \( 4x + 6 \).