Answer
$$A = - \frac{2}{7}$$
Work Step by Step
$$\eqalign{
& 3x - 2 = A\left( {3{x^2} + 1} \right) + \left( {Bx + C} \right)\left( {x + 4} \right) \cr
& {\text{Let }}x = - 4 \cr
& 3\left( { - 4} \right) - 2 = A\left( {3{{\left( { - 4} \right)}^2} + 1} \right) + \left( {B\left( { - 4} \right) + C} \right)\left( { - 4 + 4} \right) \cr
& - 12 - 2 = A\left( {49} \right) + \left( { - 4B + C} \right)\left( 0 \right) \cr
& - 14 = A\left( {49} \right) + 0 \cr
& A = - \frac{{14}}{{49}} \cr
& A = - \frac{2}{7} \cr} $$
