If `alpha, beta` are the roots of the equationn `x^(2)-3x+5=0`
and `gamma, delta` are the roots of the equation `x^(2)+5x-3=0`, then the equation whose roots are `alpha gamma+beta delta` and `alpha delta+beta gamma` is

Options

A. `x^(2)-15x-158=0`
B. `x^(2)+15x-158=0`
C. `x^(2)-15x+158=0`
D. `x^(2)+15x+158=0`

Explanation

`:'alpha+beta=3, alpha beta=5, gamma +delta=(-5),gamma delta=(-3)`
Sum of roots `=(alpha gamma +beta delta)=(alpha delta +beta gamma)`
`=(alpha +beta)(gamma+delta)=3xx(-5)=(-15)`
Product of roots `=(alpha gamma+beta delta)(alpha delta +beta gamma)`
`=alpha^(2) gama delta +alpha beta gamma^(2)+beta alpha delta^(2)+beta^(2) gamma delta`
`=gamma delta(alpha^(2)+beta^(2))+alpha beta (gamma^(2)+delta^(2))`
`=-3(alpha^(2)+beta^(2))+5(gamma^(2)+delta^(2))`
`=-3[(alpha+beta)^(2)-2alpha beta]+5[(gamma+delta)^(2)-2gamma delta]`
`=-3[9-10]+5[25+6]=158`
`:.` Required equation is `x^(2)+15x+158=0`