Albegra 2, how do I divide this

x^2 + 2x - 1 / x + 3

meaning

x^2 + 2x - 1 divided by x+3

A 4-step iterative process to divide polynomials
(iterative = fancy word for "repeated loop")

(x^2 +2x - 1) / (x + 3)

Step 1)
Take the highest degree term in the numerator (x^2) and divide it by the highest-degree term in the denominator (x):
x^2 / x = x

Step 2)
Take this result and multiply it by the denominator:
x(x + 3) = x^2 + 3x

Step 3)
Subtract this rest from the numerator (careful with signs):
(x^2 + 2x - 1) - (x^2 + 3x) = -x - 1

Step 4)
Summarize what you have so far:
(x^2 + 2x - 1)/(x+3) = x with a remainder of -x - 1

Repeat the 4 steps, using the remainder as the new numerator:

(-x - 1) / (x + 3)

Step 1) -x/x = -1
Step 2) -1(x+3) = -x - 3
Step 3) -x - 1 + x + 3 = +2
Step 4)
(x^2 + 2x - 1)/(x + 3) = x - 1 with a remainder of 2

Which can be rewritten as

x - 1 + 2/(x+3)

(only the 2 is divided by (x+3) )

x^2 + 2x - 1 / x + 3

Factor the numerator to see if we can get a factor x + 3

won't factor into integers factor

Use synthetic division to get the result and remainder.

(x + 3)/(x^2 + 2x -1) = x^2 + 2z -1 - x^2 - 3x = x - 1

(x-3)/x-1 = 2

so the result is x - 1 - 1 = x - 2 rem 2.