Answer
$A = \frac{4}{3}; B = \frac{2}{3}$
Work Step by Step
The given equation is:
$$\dfrac{2x}{(x+2)(x-1)} =\dfrac{A}{(x+2)}+\dfrac{B}{(x-1)} ~~~(1)$$
On multiplying equation (1) with $(x+2) (x-1)$, we obtain:
$$2x =A(x-1)+B(x+2)~~~(2)$$
Now, we will put $x=1$ in equation (2) to eliminate $A$ as follows:
$$2(1) =A(1-1)+B(1+2)\\ B=\dfrac{2}{3}$$
Next, we will put $x=-2;$ in equation (2) to eliminate $B$ as follows:
$$2(-2) =A(-2-1)+B(-2+2)\\ A=\dfrac{4}{3}$$
Solve equations (3) and (4) to obtain:
$$A = \dfrac{4}{3}; B = \dfrac{2}{3}$$
