Divide the polynomial 4x2+4x−10 by 2x−3 to determine the quotient and remainder.(1 point) Responses The quotient is 2x+5 . The remainder is −5 The quotient is 2 x plus 5. The remainder is negative 5 The quotient is 2x−5 . The remainder is 25 The quotient is 2 x minus 5. The remainder is 25 The quotient is 2x+5 . The remainder is 5 The quotient is 2 x plus 5. The remainder is 5 The quotient is 2x−5 . The remainder is −25

Answer 1

To divide the polynomial \( 4x^2 + 4x - 10 \) by \( 2x - 3 \), we will use polynomial long division.

1. Divide the leading term of the numerator by the leading term of the denominator: \[ \frac{4x^2}{2x} = 2x \]

2. Multiply the entire divisor by this result: \[ 2x(2x - 3) = 4x^2 - 6x \]

3. Subtract this from the original polynomial: \[ (4x^2 + 4x - 10) - (4x^2 - 6x) = 4x + 6x - 10 = 10x - 10 \]

4. Now, divide the leading term of the new polynomial by the leading term of the divisor: \[ \frac{10x}{2x} = 5 \]

5. Multiply the entire divisor by this result: \[ 5(2x - 3) = 10x - 15 \]

6. Subtract this from the polynomial: \[ (10x - 10) - (10x - 15) = -10 + 15 = 5 \]

At this point, we have finished our division process. The quotient is \( 2x + 5 \) and the remainder is \( 5 \).

Thus, the answer is:

• The quotient is \( 2x + 5 \).
• The remainder is \( 5 \).

So the correct response is: The quotient is 2x + 5. The remainder is 5.