Question 1074598
Rational coefficients means that complex roots come in complex conjugate pairs.
{{{f(x)=a(x-4)(x-2i)(x+2i)}}}
{{{f(x)=a(x-4)(x^2+4)}}}
So,
{{{f(1)=a(1-4)(1^2+4)=30}}}
{{{a(-3)(5)=30}}}
{{{-15a=30}}}
{{{a=-2}}}
So,
{{{f(x)=-2(x-4)(x^2+4)}}}
{{{f(x)=-2x^3+8x^2-8x+32}}}