First of all, we don't have a formula for solving polynomials with degree 4. We can try to use the rational root theorem, which states that if this polynomial has a rational root, it must be a divisor of 16.
So, we have to try [tex]\pm1,\ \pm2,\ \pm4,\ \pm8,\ \pm16[/tex]
You will see that 4 is a root. So, we can divide
[tex]\dfrac{x^4-8x^3+17x^2-8x+16}{x-4}=x^3-4x^2+x-4[/tex]
We can repeat the process, and check all the divisors of -4, i.e. [tex]\pm1,\ \pm2,\ \pm4[/tex]
You'll see that 4 is again a root, and we have
[tex]\dfrac{x^3-4x^2+x-4}{x-4}=x^2+1[/tex]
Now we have a quadratic polynomial which we can simply solve:
[tex]x^2+1=0 \iff x^2=-1 \iff x=\pm i[/tex]
So, the roots of this polynomials are 4 (twice), i and -i.
