1. **State the problem:** Simplify the complex expression $$z = \frac{(4 - 6i)(2 + i)}{(3 + i)(1 + i)}$$ into the form $$a + ib$$ where $$i = \sqrt{-1}$$, and find the magnitude $$|z|$$. 2. **Recall formulas and rules:** - To multiply complex numbers, use distributive property: $$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$. - To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. - The conjugate of $$x + yi$$ is $$x - yi$$. - The magnitude of $$z = a + bi$$ is $$|z| = \sqrt{a^2 + b^2}$$. 3. **Simplify numerator:** $$(4 - 6i)(2 + i) = 4 \times 2 + 4 \times i - 6i \times 2 - 6i \times i = 8 + 4i - 12i - 6i^2$$ Since $$i^2 = -1$$, this becomes: $$8 + 4i - 12i - 6(-1) = 8 - 8i + 6 = 14 - 8i$$ 4. **Simplify denominator:** $$(3 + i)(1 + i) = 3 \times 1 + 3 \times i + i \times 1 + i \times i = 3 + 3i + i + i^2 = 3 + 4i - 1 = 2 + 4i$$ 5. **Multiply numerator and denominator by conjugate of denominator:** Conjugate of denominator $$2 + 4i$$ is $$2 - 4i$$. $$z = \frac{14 - 8i}{2 + 4i} \times \frac{2 - 4i}{2 - 4i} = \frac{(14 - 8i)(2 - 4i)}{(2 + 4i)(2 - 4i)}$$ 6. **Calculate numerator:** $$(14 - 8i)(2 - 4i) = 14 \times 2 - 14 \times 4i - 8i \times 2 + 8i \times 4i = 28 - 56i - 16i + 32i^2$$ Since $$i^2 = -1$$: $$28 - 72i + 32(-1) = 28 - 72i - 32 = -4 - 72i$$ 7. **Calculate denominator:** $$(2 + 4i)(2 - 4i) = 2^2 - (4i)^2 = 4 - 16i^2 = 4 - 16(-1) = 4 + 16 = 20$$ 8. **Write fraction:** $$z = \frac{-4 - 72i}{20}$$ 9. **Simplify by dividing numerator and denominator by 4:** $$z = \frac{\cancel{4}(-1 - 18i)}{\cancel{4}5} = \frac{-1 - 18i}{5} = -\frac{1}{5} - \frac{18}{5}i$$ 10. **Final form:** $$z = -0.2 - 3.6i$$ 11. **Calculate magnitude:** $$|z| = \sqrt{\left(-\frac{1}{5}\right)^2 + \left(-\frac{18}{5}\right)^2} = \sqrt{\frac{1}{25} + \frac{324}{25}} = \sqrt{\frac{325}{25}} = \sqrt{13}$$ **Answer:** $$z = -\frac{1}{5} - \frac{18}{5}i$$ and $$|z| = \sqrt{13}$$.