If `alpha` and `beta` be the roots of the equation `x^(2)+3x+1=0`, then the value of `((alpha)/(1+beta))^(2)+((beta)/(alpha+1))^(2)` is equal to

Options

A. `15`
B. `18`
C. `21`
D. `17`

Answer

B

Explanation

`x^(2) + 3 x + 1 = 0`
Let roots be `alpha` & `beta`
`alpha + beta = -3`
`alphabeta = 1`
Also `alpha^(2) = -3alpha - 1`
`beta^(2) = -3beta -1`
`Sum = (alpha^(2))/((l+beta)^(2)) + (beta^(2))/(l +alpha)^(2)`
`=((-3alpha - 1)/(-beta)) + ((-3alpha -1)/(-alpha))`
`= (alpha(1+3alpha)+beta(1+3beta))/(alphabeta)`
`= 3(alpha^(2) + beta^(2)) + (alpha + beta)`
`= 3[9-2]+ (-3)`
`= 21 - 3`
`= 18`