1. **State the problem:** Simplify the expression $$\frac{1 - i}{4 + 4i}$$ and write the answer in the form $a + bi$ with reduced fractions. 2. **Recall the formula:** To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $4 + 4i$ is $4 - 4i$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{1 - i}{4 + 4i} \times \frac{4 - 4i}{4 - 4i} = \frac{(1 - i)(4 - 4i)}{(4 + 4i)(4 - 4i)}$$ 4. **Expand numerator:** $$(1)(4) + (1)(-4i) + (-i)(4) + (-i)(-4i) = 4 - 4i - 4i + 4i^2$$ 5. **Simplify numerator using $i^2 = -1$:** $$4 - 8i + 4(-1) = 4 - 8i - 4 = 0 - 8i = -8i$$ 6. **Expand denominator:** $$(4)(4) - (4)(4i) + (4i)(4) - (4i)(4i) = 16 - 16i + 16i - 16i^2$$ 7. **Simplify denominator:** $$16 - 16i + 16i - 16(-1) = 16 + 16 = 32$$ 8. **Write the fraction:** $$\frac{-8i}{32}$$ 9. **Simplify the fraction by dividing numerator and denominator by 8:** $$\frac{\cancel{-8}i}{\cancel{32}} = \frac{-1i}{4} = -\frac{1}{4}i$$ 10. **Final answer in $a + bi$ form:** $$0 - \frac{1}{4}i$$ So, the simplified form is $0 - \frac{1}{4}i$ or simply $-\frac{1}{4}i$.