c. Simplify \( \frac{4 x}{3 x^{2}+19 x-14}+\frac{1}{3 x^{2}+19 x-14}-\frac{x+3}{3 x^{2}+19 x-14} \)

To simplify the expression \[ \frac{4 x}{3 x^{2}+19 x-14}+\frac{1}{3 x^{2}+19 x-14}-\frac{x+3}{3 x^{2}+19 x-14}, \] we first notice that all the fractions have the same denominator, \(3 x^2 + 19 x - 14\). Therefore, we can combine the numerators: \[ \frac{4 x + 1 - (x + 3)}{3 x^2 + 19 x - 14}. \] Now simplify the numerator: \[ 4x + 1 - x - 3 = (4x - x) + (1 - 3) = 3x - 2. \] This gives us: \[ \frac{3x - 2}{3 x^2 + 19 x - 14}. \] Next, we can factor the quadratic \(3 x^2 + 19 x - 14\) if possible. We look for two numbers that multiply to \(3 \times -14 = -42\) and add up to \(19\). These numbers are \(21\) and \(-2\). Therefore, we can factor the quadratic as follows: \[ 3 x^2 + 21 x - 2 x - 14 = 3x(x + 7) - 2(x + 7) = (3x - 2)(x + 7). \] Now we rewrite our expression: \[ \frac{3x - 2}{(3x - 2)(x + 7)}. \] We can cancel \(3x - 2\) from the numerator and denominator, which gives us: \[ \frac{1}{x + 7} \quad \text{(as long as \(3x - 2 \neq 0\), which means \(x \neq \frac{2}{3}\))}. \] Thus, the simplified expression is: \[ \frac{1}{x + 7} \quad \text{for } x \neq \frac{2}{3}. \]