91Ó°ÊÓ

First, divide the leading terms. Such that: \(y\) is obtained by dividing \(4y^{2}\) by \(2y\). Write down the \(y\) above the expression. Then multiply \((2y+1)\) by \(y\) (the obtained quotient) and subtract the result from the original polynomial. More specifically: Subtract \(2y^{2} + y\) (which is \(y(2y+1)\)) from \(4y^{2}-8y\) gives \(2y^2 -9y\). Compare new formed polynomial \(2y^2 - 9y -5\) with \(2y+1\). Divide the leading term of the new polynomial by the leading term of the divisor, \(2y^{2}\) divided by \(2y\) is \(y\). Write down this result under the previously write \(y\), then perform mentioned multiply and subtract, eventually obtain \(-10y -5\). Compare \(-10y -5\) with \(2y+1\). Divide \(-10\) by \(2\), result in \(-5\), write this result under the previous written number. Finally, multiply \((2y+1)\) by \(-5\) and subtract from \(-10y -5\), get 0 as remainder. Therefore, the quotient is \(y -5\).

Check the answer using the relation \[dividend = divisor \times quotient + remainder\]. Substitute the divisor (\(2y+1\)), quotient (\(y -5\)), and remainder (0) into the equation, one can obtain the original dividend \((4y^{2}-8y-5)\). Hence the answer is correct.