1. We are given a polynomial defined by its roots: $1 - 7i$, $1 + 7i$, $4i\sqrt{3}$, and $-4i\sqrt{3}$. We want to find the explicit form of the polynomial $f(x)$. 2. Recall that if $r$ is a root, then $(x - r)$ is a factor. Hence, $$f(x) = (x - (1 - 7i))(x - (1 + 7i))(x - 4i\sqrt{3})(x + 4i\sqrt{3}).$$ 3. Group conjugate pairs together to simplify: $$(x - (1 - 7i))(x - (1 + 7i))$$ and $$(x - 4i\sqrt{3})(x + 4i\sqrt{3}).$$ 4. Expand the first pair: $$egin{aligned} (x - (1 - 7i))(x - (1 + 7i)) &= (x - 1 + 7i)(x - 1 - 7i) \\ &= [(x-1) + 7i][(x-1) - 7i] \\ &= (x - 1)^2 - (7i)^2 \\ &= (x - 1)^2 - (-49) \\ &= (x - 1)^2 + 49 \\ &= x^2 - 2x + 1 + 49 \\ &= x^2 - 2x + 50. \end{aligned}$$ 5. Expand the second pair: $$egin{aligned} (x - 4i\sqrt{3})(x + 4i\sqrt{3}) &= x^2 - (4i\sqrt{3})^2 \\ &= x^2 - (16 \cdot (-1) \cdot 3) \\ &= x^2 + 48. \end{aligned}$$ 6. Now multiply the two quadratic expressions: $$f(x) = (x^2 - 2x + 50)(x^2 + 48).$$ 7. Expand: $$egin{aligned} f(x) &= x^2 \cdot x^2 + x^2 \cdot 48 - 2x \cdot x^2 - 2x \cdot 48 + 50 \cdot x^2 + 50 \cdot 48 \\ &= x^4 + 48x^2 - 2x^3 - 96x + 50x^2 + 2400 \\ &= x^4 - 2x^3 + (48x^2 + 50x^2) - 96x + 2400 \\ &= x^4 - 2x^3 + 98x^2 - 96x + 2400. \end{aligned}$$ **Final answer:** $$f(x) = x^4 - 2x^3 + 98x^2 - 96x + 2400.$$ This polynomial has the given roots, confirming correctness.