a) $(1 - i)^2$
\[(1 - i)^2 = (1 - i)(1 - i)\]
\[= 1 \cdot 1 - 1 \cdot i - i \cdot 1 + i^2\]
\[= 1 - i - i + (-1)\]
\[= (1 - 1) - 2i = -2i\]
✅ Result: $-2i$
c) $(2 + 3i)^2$
\[(2 + 3i)^2 = (2 + 3i)(2 + 3i)\]
\[= 2\cdot 2 + 2 \cdot 3i + 3i \cdot 2 + (3i)^2\]
\[= 4 + 6i + 6i + 9i^2\]
\[= 4 + 12i + 9(-1)\]
\[= 4 + 12i - 9\]
\[= -5 + 12i\]
✅ Result: $-5 + 12i$
e) $(2 - i)(3 - 4i)$
\[(2 - i)(3 - 4i) = 2 \cdot 3 + 2 \cdot (-4i) + (-i)\cdot 3 + (-i)(-4i)\]
\[= 6 - 8i - 3i + 4i^2\]
\[= 6 - 11i + 4(-1)\]
\[= 6 - 11i - 4\]
\[= 2 - 11i\]
✅ Result: $2 - 11i$
✨ Final Answers:
- a) $-2i$
- c) $-5 + 12i$
- e) $2 - 11i$