(-4+i)(3-2i)+(2+i)=0

Solution for (-4+i)(3-2i)+(2+i)=0 equation:

Simplifying
(-4 + i)(3 + -2i) + (2 + i) = 0

Multiply (-4 + i) * (3 + -2i)
(-4(3 + -2i) + i(3 + -2i)) + (2 + i) = 0
((3 * -4 + -2i * -4) + i(3 + -2i)) + (2 + i) = 0
((-12 + 8i) + i(3 + -2i)) + (2 + i) = 0
(-12 + 8i + (3 * i + -2i * i)) + (2 + i) = 0
(-12 + 8i + (3i + -2i2)) + (2 + i) = 0

Combine like terms: 8i + 3i = 11i
(-12 + 11i + -2i2) + (2 + i) = 0

Remove parenthesis around (2 + i)
-12 + 11i + -2i2 + 2 + i = 0

Reorder the terms:
-12 + 2 + 11i + i + -2i2 = 0

Combine like terms: -12 + 2 = -10
-10 + 11i + i + -2i2 = 0

Combine like terms: 11i + i = 12i
-10 + 12i + -2i2 = 0

Solving
-10 + 12i + -2i2 = 0

Solving for variable 'i'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + 6i + -1i2) = 0

Factor a trinomial.
2((-5 + i)(1 + -1i)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-5 + i)' equal to zero and attempt to solve:

Simplifying
-5 + i = 0

Solving
-5 + i = 0

Move all terms containing i to the left, all other terms to the right.

Add '5' to each side of the equation.
-5 + 5 + i = 0 + 5

Combine like terms: -5 + 5 = 0
0 + i = 0 + 5
i = 0 + 5

Combine like terms: 0 + 5 = 5
i = 5

Simplifying
i = 5
Subproblem 2
Set the factor '(1 + -1i)' equal to zero and attempt to solve:

Simplifying
1 + -1i = 0

Solving
1 + -1i = 0

Move all terms containing i to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + -1i = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -1i = 0 + -1
-1i = 0 + -1

Combine like terms: 0 + -1 = -1
-1i = -1

Divide each side by '-1'.
i = 1

Simplifying
i = 1
Solution
i = {5, 1}