You are given a polynomial f, where f(x)=x^4 - 14x^3 + 74 x^2 -184x + 208, you are told that f(5+i)=0. Express f as the product of two quadratic polynomials and state all roots of f.

Since x=5+i is a solution to f(x)=0 we then know that x=5-i must also be a solution to f(x)=0, by the complex conjugate root theorem.Now we can break f down into the product of a polynomial and these two known roots;f(x)=(x-(5+i))(x-(5-i))p(x), where p(x) is to be found. Expanding brackets then gives us that;f(x)=(x^2-10x+26)p(x) We can then divide f(x) by (x-10x+26) to find p(x), and hence express f as the product of two quadratic polynomials. f(x)=(x^2-10x+26)(x^2-4x+8)Then by using the quadratic equation we can find the roots of p(x) and so now we have the roots of f as required. x= 5+i, 5-i, 2-2i, 2+2i