Question Use the long division method to find the result when 2x^3+7x^2+14x+12 is divided by 2x+3 Answer Attemptiout of 2 square

To divide the polynomial $2x^{3}+7x^{2}+14x+12$ by $2x+3$ using the long division method, we follow these steps:<br /><br />1. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.<br />2. Multiply the divisor by the first term of the quotient and subtract the result from the dividend to get a new dividend.<br />3. Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor.<br /><br />Let's perform the long division:<br /><br />```<br /> x^2 + 2x + 4<br /> -------------------<br />2x + 3 | 2x^3 + 7x^2 + 14x + 12<br /> - (2x^3 + 3x^2)<br /> ----------------<br /> 4x^2 + 14x + 12<br />```<br /><br />Now, we divide the leading term of the new dividend, $4x^2$, by the leading term of the divisor, $2x$, to get the next term of the quotient, $2x$.<br /><br />```<br /> x^2 + 2x + 4<br /> -------------------<br />2x + 3 | 2x^3 + 7x^2 + 14x + 12<br /> - (2x^3 + 3x^2)<br /> ----------------<br /> 4x^2 + 14x + 12<br /> - (4x^2 + 6x)<br /> ----------------<br /> 8x + 12<br />```<br /><br />Next, we divide the leading term of the new dividend, $8x$, by the leading term of the divisor, $2x$, to get the next term of the quotient, $4$.<br /><br />```<br /> x^2 + 2x + 4<br /> -------------------<br />2x + 3 | 2x^3 + 7x^2 + 14x + 12<br /> - (2x^3 + 3x^2)<br /> ----------------<br /> 4x^2 + 14x + 12<br /> - (4x^2 + 6x)<br /> ----------------<br /> 8x + 12<br /> - (8x + 12)<br /> ----------------<br /> 0<br />```<br /><br />The remainder is 0, which means that the division is exact. Therefore, the result of dividing $2x^{3}+7x^{2}+14x+12$ by $2x+3$ using the long division method is $x^2 + 2x + 4$.