Question

How do you simplify `(p+4)/(p^2+6p+8)`?

Answer 1

`= (1/(p+2))`

Explanation:

`(p + 4) / (p^2 + 6p + 8)`
`= (p+4)/((p+2)(p+4))`
`= (1/(p+2))`

Answer 2

`1/(p+2)`

Explanation:

Step A: `\bb(\text(Simplify the denominator.))`

1. It's in the form "`ap^2+bp+c`," which means your factors `p_1` and `p_2` must add to be `b` and multiply to be `a\cdotc`.
   In other words: `\bb(p_1+p_2=b)` and `\bb(p_1(p_2)=a\cdotc)`

2. `a=1`, `b=6`, `c=8`; so `p_1+p_2=6` and `p_1(p_2)=1\cdot8=8`
   The only two factors that fulfill these requirements are `\bb2` and `\bb4`. Which means you can factor the polynomial `p^2+6p+8` to `(p+2)(p+4)`.

3. Your new expression is `(p+4)/((p+2)(p+4))`.
   See anything you can cross out?

Step B: `\bb(\text(Identify similar terms to cancel.))`

1. We see that `p+4` occurs twice, one in the numerator, and one in the denominator.
   `(\color(red)(p+4))/((p+2)(\color(red)(p+4))`

2. You can cancel out similar terms, so we cross out those two.
   `\cancel(\color(red)(p+4))/((p+2)(\cancel(\color(red)(p+4)))`

3. Remove the crossed out parts...
   `1/((p+2)(1))`

Step C: `\text(Remove the 1s, and)\bb(\text( you have your answer!))`
`1/(p+2)`