Answer
$$-2q-10+\frac{22}{2q+1}$$
Work Step by Step
In order to match the first term of the dividend, $-4q^2$, the divisor $2q+1$ must be multiplied by $-2q$. The $-2q$ will go on top of your division sign.
$$(-2q)(2q+1) = -4q^2-2q$$
Then, subtract this from the first two terms of the dividend.
$$(-4q^2-22q+12)-(-4q^2-2q)=-20q$$
Bring down the next term of the dividend.
$$-20q+12$$
Multiply the divisor by $-10$ to match this. The $-10$ will follow the $-2q$ on top of your division sign.
$$(-10)(2q+1) = -20q-10$$
Subtract this from the $-20q+12$
$$(-20q+12)-(-20q-10) = 22$$
Since your remainder is $22$, you can add $\frac{22}{2q+1}$ to your answer.
$$-2q-10+\frac{22}{2q+1}$$
