Using long division method, divide the polynomial $4{p^3} - 12{p^2} - 37p - 15$ by $2p + 1$
A) $2{p^2} - 7p - 15$
B) $2{p^2} + 7p - 15$
C) $2{p^2} - 7p + 15$
D) $2{p^2} + 7p + 15$

Hint: The given cubic polynomial $4{p^3} - 12{p^2} - 37p - 15$ is divided by linear polynomial $2p + 1$so we get the quotient as a quadratic polynomial. It is given to divide the polynomial by a long division method so our basic aim is to cancel the highest power term in the first step then the second highest power term in the second step then proceed continuously until the order of remainder is less than the order of divisor. In this case order of remainder will be zero.

Complete step-by-step answer:
Here, the given cubic polynomial is $4{p^3} - 12{p^2} - 37p - 15$. It is to be divided by a linear polynomial $2p + 1$ which is called as a divisor.
Steps of long division
Step1: first multiply the divisor by a suitable number such that product highest power term is equal to the highest term of given polynomials.
Step2: subtract the product from the given polynomial by changing sign of product terms as shown below.
Step3: then bring down the next term and apply the step 1 to cancel the second highest power terms.
Step4: repeat first two steps until the order of remainder is less than the order of the divisor.
Now,
$
  \left. {2p + 1} \right)4{p^3} - 12{p^2} - 37p - 15\left( {2{p^2} - 7p - 15} \right. \\
  \,\,\,\,\,\,\,\,\,\,\,\,\underline { \pm 4{p^3} \pm 2{p^2}} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\, - 14{p^2} - 37p \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { \mp 14{p^2} \mp 7p} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\, - 30p - 15 \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { \mp 30p \mp 15} \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 + \,\,\,\,\,0 \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
 $
Thus, the quotient for the division of a given polynomial is $2{p^2} - 7p - 15$.

Hence, the correct option is (A).

Note: This question is also solved by the factor method. To divide by factor method we have to first factorise the given polynomial and then divide the factors by divisor and we get that divisor cancel one of the factor of the polynomial and the remaining factors will be the quotient for the division of the given polynomials.
This method is applicable only for the polynomials which are completely divisible by the divisor polynomial.