Answer
$x^5-12x^4+74x^3-248x^2+445x-500$
Work Step by Step
Step 1. Given zeros $x=4, 1-2i, 3+4i$, we can identify two more zeros as $x=1+2i, 3-4i$
Step 2. We can write the polynomial as $f(x)=(x-4)(x-1+2i)(x-1-2i)(x-3-4i)(x-3+4i)=(x-4)((x-1)^2-(2i)^2)((x-3)^2-(4i)^2)=(x-4)(x^2-2x+1+4)(x^2-6x+9+16)=(x-4)(x^2-2x+5)(x^2-6x+25)$
Step 3. Continue from above, we have $f(x)=(x-4)(x^4+(-6-2)x^3+(25+5+12)x^2+(-50-30)x+125)=(x-4)(x^4-8x^3+42x^2-80x+125)=x^5-12x^4+74x^3-248x^2+445x-500$