1. **Problem:** Given complex numbers $z_1 = 1 + i$, $z_2 = 3 - 2i$, and $z_3 = -2 + 3i$, calculate $z_1 \times z_2 + 2z_3^*$. 2. **Formula and rules:** - Multiplication of complex numbers: $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$. - Conjugate of $z = a + bi$ is $z^* = a - bi$. - Addition of complex numbers: add real parts and imaginary parts separately. 3. **Calculate $z_1 \times z_2$:** $$z_1 z_2 = (1 + i)(3 - 2i) = (1 \times 3 - 1 \times (-2)) + (1 \times (-2) + 1 \times 3)i = (3 + 2) + (-2 + 3)i = 5 + i$$ 4. **Calculate $2z_3^*$:** Conjugate $z_3^* = -2 - 3i$ $$2z_3^* = 2(-2 - 3i) = -4 - 6i$$ 5. **Add results:** $$z_1 z_2 + 2z_3^* = (5 + i) + (-4 - 6i) = (5 - 4) + (1 - 6)i = 1 - 5i$$ **Final answer:** $1 - 5i$