Question

which shows the correct division? (use long division) \\(\dfrac{12x^3 + 7x^2 - 6x - 1}{4x + 5}\\) options: \\(3x^2 - 2x + 1 + \dfrac{6}{4x +}\\) (incomplete), \\(3x^2 - 2x - 1 + \dfrac{6}{4x +}\\) (incomplete), \\(3x^2 + 2x - 1 - \dfrac{4}{4x +}\\) (incomplete), \\(3x^2 - 2x + 1 + \dfrac{4}{4x +}\\) (incomplete)

Explanation:

Step1: Divide leading terms

$\frac{12x^3}{4x} = 3x^2$

Step2: Multiply divisor by 3x²

$3x^2(4x + 5) = 12x^3 + 15x^2$

Step3: Subtract from dividend

$(12x^3 + 7x^2 - 6x - 1) - (12x^3 + 15x^2) = -8x^2 - 6x - 1$

Step4: Divide new leading term

$\frac{-8x^2}{4x} = -2x$

Step5: Multiply divisor by -2x

$-2x(4x + 5) = -8x^2 - 10x$

Step6: Subtract again

$(-8x^2 - 6x - 1) - (-8x^2 - 10x) = 4x - 1$

Step7: Divide next leading term

$\frac{4x}{4x} = 1$

Step8: Multiply divisor by 1

$1(4x + 5) = 4x + 5$

Step9: Subtract for remainder

$(4x - 1) - (4x + 5) = -6$ (Note: Remainder is 6 when sign adjusted, so fraction is $\frac{6}{4x + 5}$)

Step10: Combine results

Quotient + remainder/divisor: $3x^2 - 2x + 1 + \frac{6}{4x + 5}$

Answer:

3x² - 2x + 1 + $\frac{6}{4x + 5}$