Find the quotient and remainder after  2x^2+7x+32x 2 +7x+3  is divided by  x+6x+6  using long division method.

It seems like there's a bit of confusion in your question as the expressions are not clear. However, I'll assume you want to divide the polynomial 2x^2 + 7x + 3 by x + 6. Here's how you can do it:

Step 1: Write the division in long division format.

        2x^2 + 7x + 3
_______________________

x + 6 |

Step 2: Divide the first term of the numerator by the first term of the denominator. This gives you the first term of the quotient.

        2x
_______________________

x + 6 | 2x^2 + 7x + 3

Step 3: Multiply the entire denominator by the first term of the quotient and subtract this from the original numerator. This gives you the new numerator.

        2x
_______________________

x + 6 | 2x^2 + 7x + 3 - (2x^2 + 12x) _______________ -5x + 3

Step 4: Repeat the process with the new numerator. Divide the first term of the new numerator by the first term of the denominator. This gives you the next term of the quotient.

        2x - 5
_______________________

x + 6 | 2x^2 + 7x + 3 - (2x^2 + 12x) _______________ -5x + 3 - (-5x + 30) ___________ -27

So, the quotient is 2x - 5 and the remainder is -27.