if x^2+px+1 is a factor of expression ax^3+bx^2+c then prove that a^2_c^2=_bc

ax³+bx²+c=(x²+px+c)(kx+m) where kx+m is a linear factor

expandinding the above and comparing coefficients of same powers of x we get

p=-a/c,ap+c=b,put p=-a/c in ap+c=b

we get -a²/c+c=b

-a²+c²=bc

so a²-c²=-bc

hence proved

suppose (ex+f) is the other root. Therefore,