Introduction:
In this problem, we are given a polynomial (x4 2x3-2x2 4x 6) and we need to find the polynomial that must be subtracted from it so that the result is exactly divisible by (x2 2x-3).

Solution:
To find the polynomial that must be subtracted from (x4 2x3-2x2 4x 6), we will use the long division method. The steps are as follows:

Step 1: Write the dividend and divisor in long division format.

x2 + 2x - 3 | x4 + 2x3 - 2x2 + 4x + 6

Step 2: Divide the first term of the dividend by the first term of the divisor and write the result on top.

x2 + 2x - 3 | x4 + 2x3 - 2x2 + 4x + 6
x2

Step 3: Multiply the divisor by the quotient obtained in step 2 and subtract the result from the dividend.

x2 + 2x - 3 | x4 + 2x3 - 2x2 + 4x + 6
x2 x4 + 2x3 - 3x2
--------------
x3 - 3x2 + 4x

Step 4: Repeat steps 2 and 3 with the remainder obtained in step 3.

x2 + 2x - 3 | x4 + 2x3 - 2x2 + 4x + 6
x2 x4 + 2x3 - 3x2
--------------
x3 - 3x2 + 4x
x3 + 2x2 - 3x
------------
-5x + 6

Step 5: The remainder (-5x + 6) is the polynomial that must be subtracted from (x4 2x3-2x2 4x 6) so that the result is exactly divisible by (x2 2x-3).

Therefore, (-5x + 6) must be subtracted from (x4 2x3-2x2 4x 6) so that the result is exactly divisible by (x2 2x-3).

Conclusion:
To find the polynomial that must be subtracted from (x4 2x3-2x2 4x 6) so that the result is exactly divisible by (x2 2x-3), we used the long division method. The remainder obtained from the long division (-5x + 6) is the polynomial that must be subtracted from (x4 2x3-2x2 4x 6).

(2x + 9) should be subtracted