f(x)=x^4+ax^3+bx^2+cx+d and has real coefficients. If two of the roots are 4−i and 2+3i, what is the value of a+b+c+d?

Answer 1

complex roots always come in conjugate pairs. So,

f(x) = (x-(4-i))(x-(4+i))(x-(2+3i))(x-(2-3i))
= ((x-4)^2 + 1^2)((x-2)^2 + 3^2)
= (x^2-8x+17)(x^2-4x+13)
= x^4 - 12x^3 + 62x^2 - 172x + 221
...