Final answer:
To divide (x³ + 5x² - 12x - 36) by (x + 2) using long division, we get the quotient x² + 3x - 18 and a remainder of -24x² - 12x - 36. The dividend can be factored completely as -6(2x² + x + 6).
Step-by-step explanation:
To divide (x³ + 5x² - 12x - 36) by (x + 2) using long division, we start by dividing x³ by x, which gives us x². Next, we multiply (x + 2) by x², which gives us x³ + 2x². We then subtract this from the original expression, resulting in 3x² - 12x - 36.
Next, we divide 3x² by x, which gives us 3x. We multiply (x + 2) by 3x, which gives us 3x³ + 6x². Subtracting this from 3x² - 12x - 36, we get -18x² - 12x - 36.
Finally, we divide -18x² by x, which gives us -18x. Multiplying (x + 2) by -18x gives us -18x³ - 36x². Subtracting this from -18x² - 12x - 36, we get -24x² - 12x - 36.
After completing the long division, we get the quotient x² + 3x - 18 and a remainder of -24x² - 12x - 36. To factor the dividend completely, we can factor -24x² - 12x - 36, yielding -6(2x² + x + 6).