Final answer:
To find the quotient and remainder for the division of {x²+x-12}{x-3}, long division of polynomials is used, resulting in a quotient of x + 4 and a remainder of 0.
Step-by-step explanation:
The student asked to find the quotient and remainder for the division {x²+x-12}{x-3}. This is a polynomial division problem. To solve this, we perform long division similar to how we divide numbers, but with polynomials. First, we divide the highest degree term of the numerator (x²) by the highest degree term of the denominator (x) to get x as the first term of the quotient. Multiplying (x) by (x-3) gives us x² - 3x. We then subtract this from the original polynomial to get 4x - 12. Next, we divide 4x by x to get +4, thus adding 4 to our quotient. Multiplying 4 by (x-3) to subtract from our new polynomial gives a remainder of 0. Therefore, the quotient is x + 4 and the remainder is 0.