Question:

Step 1: Express the complex number in rectangular form. Let $z = x + iy$ where $x, y \in \mathbb{R}$. Step 2: Square both sides of the given equation to eliminate the modulus. We are given that $|z + 16| = 4|z + 1|$. Squaring both sides: $$|z + 16|^2 = 16|z + 1|^2$$ Step 3: Expand the modulus expressions using the rectangular form. Substituting $z = x + iy$: $$(x + 16)^2 + y^2 = 16[(x + 1)^2 + y^2]$$ Step 4: Expand both sides of the equation. Left side: $$x^2 + 32x + 256 + y^2$$ Right side: $$16(x^2 + 2x + 1 + y^2) = 16x^2 + 32x + 16 + 16y^2$$ Step 5: Simplify by moving all terms to one side. $$x^2 + 32x + 256 + y^2 = 16x^2 + 32x + 16 + 16y^2$$ Subtracting $32x$ from both sides: $$x^2 + 256 + y^2 = 16x^2 + 16 + 16y^2$$ Rearranging: $$256 - 16 = 16x^2 - x^2 + 16y^2 - y^2$$ $$240 = 15x^2 + 15y^2$$ Step 6: Factor and solve for $|z|^2$. $$240 = 15(x^2 + y^2)$$ $$x^2 + y^2 = 16$$ Step 7: Find the modulus of $z$. Since $|z|^2 = x^2 + y^2$: $$|z|^2 = 16$$ $$|z| = 4$$ **Final Answer:** The modulus of $z$ is $|z| = 4$, which corresponds to **Option 1**.