The determinant Δ=a2+xabacabb2+xbcacbcc2+x is divisible by

Δ=1aa3+axabaca2bb2+xbca2cbcc2+xApplying C1→C1+bC2+cC3 and taking a2+b2+c2+x common, we getΔ=1aa2+b2+c2+xaabacbb2+xbccbcc2+xApplying C2→C2−bC1 and C3→C3−cC1, we getΔ=1aa2+b2+c2+xa00bx0c0x=1aa2+b2+c2+xax2=x2a2+b2+c2+xThus ∆ is divisible by x and x2.