divide by long division.
divide x^3 + 4x^2 - 3x - 12 by x^2 - 3
This page explains the procedure quite well.
http://www.sosmath.com/algebra/factor/fac01/fac01.html
The example about 2/3 of the way down the site matches your question quite nicely.
you get x+4
To divide the polynomial x^3 + 4x^2 - 3x - 12 by the polynomial x^2 - 3 using long division, follow these steps:
Step 1: Arrange the polynomials in descending order.
x^3 + 4x^2 - 3x - 12 (dividend)
÷ x^2 - 3 (divisor)
Step 2: Divide the first term of the dividend (x^3) by the first term of the divisor (x^2).
x^3 ÷ x^2 = x
Step 3: Multiply the divisor (x^2 - 3) by the result from step 2 (x).
x * (x^2 - 3) = x^3 - 3x
Step 4: Subtract the product from step 3 (x^3 - 3x) from the dividend (x^3 + 4x^2 - 3x - 12) and bring down the next term.
(x^3 + 4x^2 - 3x - 12) - (x^3 - 3x) = 7x^2 - 12x - 12
Step 5: Repeat steps 2 to 4 using the new polynomial obtained in step 4 (7x^2 - 12x - 12).
Step 2: Divide the first term of the new polynomial (7x^2) by the first term of the divisor (x^2).
7x^2 ÷ x^2 = 7
Step 3: Multiply the divisor (x^2 - 3) by the result from step 2 (7).
7 * (x^2 - 3) = 7x^2 - 21
Step 4: Subtract the product from step 3 (7x^2 - 21) from the new polynomial (7x^2 - 12x - 12) and bring down the next term.
(7x^2 - 12x - 12) - (7x^2 - 21) = -12x + 9
Step 5: Repeat steps 2 to 4 using the new polynomial obtained in step 4 (-12x + 9).
Step 2: Divide the first term of the new polynomial (-12x) by the first term of the divisor (x^2).
-12x ÷ x^2 = -12 / x
Since the degree of the divisor is greater than the degree of the new polynomial, we cannot continue the long division process. So, the final result of the division is:
Quotient: x + 7
Remainder: -12x + 9
Therefore, the division of x^3 + 4x^2 - 3x - 12 by x^2 - 3 using long division is equal to x + 7 with a remainder of -12x + 9.