Let a, b, and c be the roots of the polynomial p(x)=x^3+2x^2+3x+4.

Let g(x) be the monic polynomial?

whose roots are a+b, a+c, and b+c (each with multiplicity 1). Determine g(x).

sum of the roots = - coefficient of x^2

hence

a + b + c = -2

So a+b = -2-c

a +c = - 2-b

b+c = - 2- a

so we need to find equation whose roots are 2-a, 2-b 2-c

p(x)=x^3+2x^2+3x+4 has roots a,b,c

so p(-x) = -x^3 + 2x^2 – 3x + 4

or f(x) = x^3 – 2x + 3x – 4 (dividing p(-x) by – 1) has roots –a , -b ,- c

so f(x+2) = (x+2)^3 – 2(x+2)^2 + 3(x+2) – 4 has roots -2 – a , -2 –b , -2 - c

so g(x) = f(x+ 2) = x(x+2)^2 + 3 x+ 6- 4

= x(x^2 + 4x+ 4)+ 3x + 6 -2

= x^3 + 4x^2 + 7x + 4 is the required result.