Answer
$\dfrac{6(x^2-x+1)}{x(2x-1)}$
Work Step by Step
By factoring both the denominator and numerator of each rational function and cancelling the common factors, we have
$$
\frac{12}{x^2+x}\cdot \frac{x^3+1}{4x-2} = \frac{12}{x(x+1)}\cdot \frac{(x+1)(x^2-x+1)}{2(2x-1)}\\
= \frac{6(x^2-x+1)}{x(2x-1)}
.$$
