Answer
$\displaystyle \frac{6x^{2}+14x+10}{(x+3)(x+4)(x+1)}$
Work Step by Step
1. Find the LCD .
1st denominator = $(x+3)(x+4)$
2nd denominator = $(x+4)(x+1)$
List factors of the 1st denominator.
From each next denominator, add only those factors that do not yet appear in the list.
LCD = $(x+3)(x+4)(x+1)$
2. Rewrite each rational expression with the the LCDas the denominator
= $\displaystyle \frac{(4x+1)(x+1)}{(x+3)(x+4)(x+1)}+\frac{(2x+3)(x+3)}{(x+4)(x+1)(x+3)}$
3. Add or subtract numerators, placing the resulting expression over the LCD.
= $\displaystyle \frac{(4x+1)(x+1)+(2x+3)(x+3)}{(x+3)(x+4)(x+1)}$
4. If possible, simplify.
= $\displaystyle \frac{4x^{2}+4x+x+1+2x^{2}+6x+3x+9}{(x+3)(x+4)(x+1)} $
= $\displaystyle \frac{6x^{2}+14x+10}{(x+3)(x+4)(x+1)}$
= $\displaystyle \frac{2(3x^{2}+7x+5)}{(x+3)(x+4)(x+1)}$
The numerator has a factor of the form $ax^{2}+bx+c.$
To factor, find factors of $ac$ whose sum is $b.$
If successful, rewrite $bx$ and factor in pairs.
... we can't find factors of $15$ whose sum is $+7$
... we can't further factor the numerator, so we leave it unfactored.
= $\displaystyle \frac{6x^{2}+14x+10}{(x+3)(x+4)(x+1)}$
