Answer:
(x+3+5i) (x+3-5i) =x^2 +6x+34
(x-4i) (x+4i) = x^2 +16
(x-6+i) (x-6-i) = x^2 -12x+37
Explanation:
(x+3+5i) (x+3-5i)
Let m = x+3
(m+5i) (m-5i)
FOIL (first outer inner last)
Remember i^2 = -1
m^2 -5mi + 5mi -25i^2 = m^2 -25(-1) = m^2 +25
What is m^2 = (x+3)^2
FOIL = x^2 +3x+3x+9 = x^2 +6x+9
Replace m^2 with x^2 +6x+9
(x+3+5i) (x+3-5i) = x^2 +6x+9+25
= x^2 +6x+34
(x-4i) (x+4i) =
FOIL = x^2 +4ix -4ix -16i^2 = x^2 -16(-1) = x^2 +16
(x-6+i) (x-6-i)
Let m = x-6
(m+i) (m-i)
m^2 -mi +mi -i^2
m^2 - (-1)
m^2+1
What m^2
(x-6) ^2 = x^2 -6x-6x+36 = x^2 -12x+36
(x-6+i) (x-6-i) = m^2 +1 =x^2 -12x+36 +1 = x^2 -12x+37