Use long division to divide the X= following. Write your remainder as a fraction. (2x^3+7x^2+2x+9)div (2x+3)

To use long division to divide \((2x^3 + 7x^2 + 2x + 9) \div (2x + 3)\), follow these steps:<br /><br />1. **Divide the leading term of the dividend by the leading term of the divisor:**<br /> \[<br /> \frac{2x^3}{2x} = x^2<br /> \]<br /> Write \(x^2\) above the division line.<br /><br />2. **Multiply the entire divisor by this result and subtract from the dividend:**<br /> \[<br /> (2x + 3) \cdot x^2 = 2x^3 + 3x^2<br /> \]<br /> Subtract this from the original dividend:<br /> \[<br /> (2x^3 + 7x^2 + 2x + 9) - (2x^3 + 3x^2) = 4x^2 + 2x + 9<br /> \]<br /><br />3. **Repeat the process with the new polynomial:**<br /> Divide the leading term of the new polynomial by the leading term of the divisor:<br /> \[<br /> \frac{4x^2}{2x} = 2x<br /> \]<br /> Write \(2x\) above the division line next to \(x^2\).<br /><br />4. **Multiply the entire divisor by this result and subtract from the new polynomial:**<br /> \[<br /> (2x + 3) \cdot 2x = 4x^2 + 6x<br /> \]<br /> Subtract this from the current polynomial:<br /> \[<br /> (4x^2 + 2x + 9) - (4x^2 + 6x) = 2x - 6x + 9 = -4x + 9<br /> \]<br /><br />5. **Continue the process:**<br /> Divide the leading term of the remaining polynomial by the leading term of the divisor:<br /> \[<br /> \frac{-4x}{2x} = -2<br /> \]<br /> Write \(-2\) above the division line next to \(x^2\) and \(2x\).<br /><br />6. **Multiply the entire divisor by this result and subtract from the remaining polynomial:**<br /> \[<br /> (2x + 3) \cdot (-2) = -4x - 6<br /> \]<br /> Subtract this from the current polynomial:<br /> \[<br /> (-4x + 9) - (-4x - 6) = -4x + 4x + 9 + 6 = 15<br /> \]<br /><br />The quotient is \(x^2 + 2x - 2\) and the remainder is \(15\). Therefore, the final answer is:<br /><br />\[<br />\boxed{x^2 + 2x - 2 \text{ with a remainder of } \frac{15}{2x + 3}}<br />\]