Question

1. $\frac{r}{2r+1} + \frac{12r-6}{4r^2-1}$
2. $\frac{2}{h+2} + \frac{5}{h+5}$

Explanation:

For Problem 11:

Step1: Factor denominator and numerator

Notice $4r^2-1=(2r-1)(2r+1)$ and $12r-6=6(2r-1)$.
The expression becomes:
$\frac{r}{2r+1} + \frac{6(2r-1)}{(2r-1)(2r+1)}$

Step2: Simplify second fraction

Cancel $(2r-1)$ (where $2r-1
eq0$):
$\frac{r}{2r+1} + \frac{6}{2r+1}$

Step3: Combine like fractions

Add numerators over common denominator:
$\frac{r + 6}{2r+1}$

Step1: Find common denominator

Common denominator is $(h+2)(h+5)$.
Rewrite each fraction:
$\frac{2(h+5)}{(h+2)(h+5)} + \frac{5(h+2)}{(h+2)(h+5)}$

Step2: Expand numerators

Calculate expanded numerators:
$\frac{2h+10}{(h+2)(h+5)} + \frac{5h+10}{(h+2)(h+5)}$

Step3: Combine and simplify

Add numerators and simplify:
$\frac{2h+10+5h+10}{(h+2)(h+5)} = \frac{7h+20}{(h+2)(h+5)}$

Answer:

$\frac{r+6}{2r+1}$ (where $r
eq -\frac{1}{2}, \frac{1}{2}$)

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For Problem 13: