Factorize each of the following quadratic polynomials by using the method of completing the square:
(1) `p^2+6p+8`
(2) `q^2-10 q+21`
(3) `p^2+6p-16`

Explanation

(1)`p^2+6p+8`
`=p^2+6p+9-1`
`=(p+3)^2-(1)^2`
Completing the square:
`=(p+3+1)(p+3-1)`
`=(p+2)(p+4)`

(2) `q^2-10q+21`
Adding and subtracting `(-5)^2`.
`=q^2+2timesqtimes(-5)+(-5)^2-(-5)^2+21`
`=(q-5)^2+21-25`
`=(q-5)^2-4`
`=(q-5)^2-2^2`
`=(q-5-2)(q-5+2)`
`=(q-7)(q-3)`

(3) `p^2+6p-16`
Adding and subtracting `(3)^2`
`=p^2+2timesptimes3+3^2-3^2-16`
`=(p+3)^2-3^2-16`
`=(p+3)^2-9-16`
`=(p+3)^2-25`
`=(p+3)^2-5^2`
`=(p+3+5)(p+3-5)`
`=(p+8)(p-2)`