If the roots $x^4+2x^3-21x^2-22+40=0$ are in A.P. then the roots are

$f\left(x\right)=x^4+2x^3-21x^2-22+40=0$

$f\left(1\right)=0\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}f(-2)=0$

$1,-2$are the two roots of the given equation 

$x^2+x-20=0$

$x^2+5x-4x-20=0$

$x(x+5)-4(x+5)=0$

$(x+5)(x-4)=0$

$x=-5,4$

$\therefore$The roots of the given equation are $1,-2,-5,4$