Answer
$$A = - 1$$
Work Step by Step
$$\eqalign{
& 3x - 1 = A{\left( {2{x^2} + 1} \right)^2} + \left( {Bx + C} \right)\left( x \right)\left( {2{x^2} + 1} \right) + \left( {Dx + E} \right)\left( x \right) \cr
& {\text{Let }}x = 0 \cr
& 3\left( 0 \right) - 1 = A{\left( {2{{\left( 0 \right)}^2} + 1} \right)^2} + \left( {B\left( 0 \right) + C} \right)\left( 0 \right)\left( {2{{\left( 0 \right)}^2} + 1} \right) + \left( {D\left( 0 \right) + E} \right)\left( 0 \right) \cr
& 0 - 1 = A{\left( {0 + 1} \right)^2} + 0 \cr
& A = - 1 \cr} $$
