1. **State the problem:** We need to simplify the expression $$(1 + 2i^2)(1 + 2i^5 + 5i^6)$$ and write it in the form $x + yi$ where $x$ and $y$ are real numbers. 2. **Recall powers of $i$:** Remember that $i$ is the imaginary unit where $i^2 = -1$. Powers of $i$ cycle every 4 steps: $$i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1,$$ and this pattern repeats. 3. **Simplify each power:** - $i^2 = -1$ - $i^5 = i^{4+1} = i^4 \cdot i = 1 \cdot i = i$ - $i^6 = i^{4+2} = i^4 \cdot i^2 = 1 \cdot (-1) = -1$ 4. **Substitute back into the expression:** $$(1 + 2(-1))(1 + 2i + 5(-1)) = (1 - 2)(1 + 2i - 5) = (-1)(-4 + 2i)$$ 5. **Multiply:** $$-1 \times (-4) = 4$$ $$-1 \times 2i = -2i$$ 6. **Write final expression:** $$4 - 2i$$ **Answer:** The expression in the form $x + yi$ is $$4 - 2i$$.