If a, b are the roots of x^2 + px + q = 0, find the value of (1) a^3.b + a.b^3 (2) a^4 + a^2.b^2 + b^4

We have the equation x^2 + px + q = 0 and given that its roots are a and b.

a + b = -p

ab = q

(1) a^3.b + a.b^3

=> ab(a^2 + b^2)

=> ab [ (a + b)^2 - 2ab ]

=> q [(-p)^2 - 2q]

=> q (p^2 - 2q)

=> p^2.q - 2q^2

(2) a^4 + a^2b^2 + b^4

=> (a^2)^2 + (a^2.b^2) + (b^2)^2

=> (a^2 + b^2)^2 - a^2.b^2

=> [(a + b)^2 - 2ab]^2 - (ab)^2

=> [(-p)^2 - 2q]^2 - q^2

=> [p^2 - 2q]^2 - q^2

=> p^4 + 4q^2 - 4p^2.q - q^2

=> p^4 + 3q^2 - 4p^2.q