Factorize the numerator.

Here are the solutions to the problems:

1. Using factorization by grouping, simplify $\frac{r^2 - k^2 + r + k}{r - k + 1}$

Step 1: Factorize the numerator.
The numerator is $r^2 - k^2 + r + k$.
We can group the terms as $(r^2 - k^2) + (r + k)$.
The term $(r^2 - k^2)$ is a difference of squares, which factors as $(r-k)(r+k)$.
So, the numerator becomes $(r-k)(r+k) + (r+k)$.
Now, factor out the common term $(r+k)$:
Numerator $= (r+k)((