Answer:
[tex](-28) + (-40\, i)[/tex].
Step-by-step explanation:
Expand the expression using the distributive property of multiplication, [tex]X \, (Y + Z) = X\, Y + X\, Z[/tex]:
[tex]\begin{aligned}& (3 + 7\, i)\, (-2 - 2\, i) - (4\, i)\, (5 - 9\, i) \\ =\; & (3 + 7\, i)\, (-2) + (3 + 7\, i)\, (-2\, i) \\ & + (-4\, i)\, (5) + (-4\, i)\, (-9\, i) \\ =\; & (3)\, (-2) + (7\, i)\, (-2) + (3)\, (-2\, i) + (7\, i)\, (-2\, i) \\ & + (-20\, i) + 36\, i^{2} \\ =\; & (-6)\, + (-14\, i) + (-6\, i)+ (-14\, i^{2}) + (-20\, i) + 36\, i^{2} \\ =\; & (-6) + (-40\, i) + 22\, i^{2}\end{aligned}[/tex].
Simplify the expression further using the fact that [tex]i^{2} = (-1)[/tex]:
[tex]\begin{aligned}& (-6) + (-40\, i) + 22\, i^{2} \\ = \; & (-6) + (-40\, i) + (-22) \\ =\; & (-28) + (-40\, i)\end{aligned}[/tex].