Question

1. find the quotient using long division. express the answer as a rational expression. \\(\overline{2x + 3)6x^2 + 5x + 8}\\) options: \\(3x - 2 + \frac{14}{2x + 3}\\), \\(3x^2 - 2x + \frac{14}{3}\\), \\(3x^2 - 2x + \frac{14}{2x + 3}\\), \\(3x - 2 + \frac{14}{3}\\)

Explanation:

Step1: Divide leading terms

$\frac{6x^2}{2x} = 3x$

Step2: Multiply divisor by $3x$

$3x(2x+3) = 6x^2 + 9x$

Step3: Subtract from dividend

$(6x^2+5x+8)-(6x^2+9x) = -4x+8$

Step4: Divide new leading terms

$\frac{-4x}{2x} = -2$

Step5: Multiply divisor by $-2$

$-2(2x+3) = -4x-6$

Step6: Subtract to get remainder

$(-4x+8)-(-4x-6) = 14$

Step7: Write final quotient

Quotient + $\frac{\text{Remainder}}{\text{Divisor}}$

Answer:

$3x - 2 + \frac{14}{2x+3}$ (matches the first option)