21factor f(a^2 + a - 3)..work backwords by assuming that f(a)|f(a^2+ a - 3)
\hspace{-16}$If $\mathbf{f(x)=x^3+x^2-4x+1}$ and If $\mathbf{\alpha}$ be a root of $\mathbf{f(x)=0}.\;$ Then Prove\\\\ that $\mathbf{\alpha^2+\alpha-3}$ is also root of $\mathbf{f(x)=0}$
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21
341a3f(a2+a-3) = (a3+a2-3a)3+a(a3+a2-3a)2-4a2(a3+a2-3a)+a3
Now if a3+a2-4a+1=0, then a3+a2-3a = a-1
Hence the given expression simplifies to
(a-1)3+a(a-1)2-4a2(a-1)+a3=-(a3+a2-4a+1)=0
Edit: we have to note that a≠0
1708Thanks hsbhatt Sir for Very Nice approach
also thanks to Shubhodip for Hint