What must be added to 8x4 14x3 – 2x2 7x – 8 so that the resulting polynomial is exactly divisible by 4x2 3x –2.

To find what must be added to the polynomial 8x^4 + 14x^3 – 2x^2 + 7x – 8 in order for it to be exactly divisible by the polynomial 4x^2 + 3x – 2, we can use polynomial long division.

Step 1: Write the dividend (8x^4 + 14x^3 – 2x^2 + 7x – 8) and the divisor (4x^2 + 3x – 2) in descending order of exponents.

Step 2: Divide the first term of the dividend (8x^4) by the first term of the divisor (4x^2) to get 2x^2. Write this as the first term of the quotient.

Step 3: Multiply the divisor (4x^2 + 3x – 2) by the first term of the quotient (2x^2) and subtract the result from the dividend (8x^4 + 14x^3 – 2x^2 + 7x – 8).

Step 4: Repeat steps 2 and 3 with the new dividend (2x^2 + 14x^3 + 7x – 8) and the divisor (4x^2 + 3x – 2).

Step 5: Continue this process until the degree of the new dividend is less than the degree of the divisor.

Step 6: The remainder obtained in the last step will be the expression that needs to be added to the original polynomial in order for it to be exactly divisible by the given polynomial.

Please note that I will now perform the polynomial long division to find the exact expression that needs to be added.