(6x^3 + 11x^2 -12x -5 ):(2x + 5 )

To divide the polynomial (6x^3 + 11x^2 - 12x - 5) by (2x + 5), we can use long division.

First, we divide the first term of the dividend (6x^3) by the first term of the divisor (2x). This gives us 3x^2. We write this as the first term of the quotient.

Next, we multiply the entire divisor (2x + 5) by the first term of the quotient (3x^2). This gives us 6x^3 + 15x^2.

We subtract this from the dividend (6x^3 + 11x^2 - 12x - 5) to get the new dividend:

(6x^3 + 11x^2 - 12x - 5) - (6x^3 + 15x^2) = -4x^2 - 12x - 5.

Now, we repeat the process with the new dividend (-4x^2 - 12x - 5) and the divisor (2x + 5).

We divide the first term of the new dividend (-4x^2) by the first term of the divisor (2x). This gives us -2x. We write this as the second term of the quotient.

Next, we multiply the entire divisor (2x + 5) by the second term of the quotient (-2x). This gives us -4x^2 - 10x.

We subtract this from the new dividend (-4x^2 - 12x - 5) to get the new dividend:

(-4x^2 - 12x - 5) - (-4x^2 - 10x) = -2x - 5.

Now, we repeat the process with the new dividend (-2x - 5) and the divisor (2x + 5).

We divide the first term of the new dividend (-2x) by the first term of the divisor (2x). This gives us -1. We write this as the third term of the quotient.

Next, we multiply the entire divisor (2x + 5) by the third term of the quotient (-1). This gives us -2x - 5.

We subtract this from the new dividend (-2x - 5) to get the new dividend:

(-2x - 5) - (-2x - 5) = 0.

Since the new dividend is 0, we have finished the division.

Therefore, the quotient is 3x^2 - 2x - 1.