Answer:
(x + 1 + i)(x + 1 - i) ↔ x²+2x+2; (x+2i)(x-2i) ↔ x²+4; (x-2i-2)(x+2i-2) ↔ x²-4x+8
Explanation:
For the first expression, use the distributive property to multiply. First multiply all terms of the second trinomial by x:
x(x)+x(1)-x(i) = x²+x-xi
Next multiply all terms of the second trinomial by 1:
1(x)+1(1)-1(i) = x+1-i
Next multiply all terms of the second trinomial by i:
i(x)+i(1)-i(i) = xi+i-i²
Combining all terms,
x²+x-xi+x+1-i+xi+i-i²
Combining like terms, we have
x²+(x+x)+(-xi+xi)+(-i+i)+1-i² = x²+2x+1-i² = x²+2x+1-(-1) = x²+2x+2
For the second expression, use the acronym FOIL (First, Outer, Inner, Last):
x(x) = x²
x(-2i) = -2xi
2i(x) = 2xi
2i(-2i) = -4i² = -4(-1) = 4
This gives us x² - 2xi + 2xi + 4, which simplifies to x²+4.
For the third expression, use the distributive property. Multiply everything in the second trinomial by x:
x(x)+x(2i)+x(-2) = x²+2xi-2x
Multiply everything in the second trinomial by -2i:
-2i(x)+-2i(2i)+-2i(-2) = -2xi-4i²+4i = -2xi-4(-1)+4i = -2xi+4+4i
Multiply everything in the third trinomial by -2:
-2(x)+-2(2i)+-2(-2) = -2x-4i+4
Combining all, we have
x²+2xi-2x+-2xi+4+4i+-2x-4i+4
Combining like terms gives us
x²+(2xi+-2xi)+(-2x+-2x)+(4+4)+(4i-4i) = x²-4x+8
