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Product Rule & Integration by Parts

Review question

Can we find the repeated root for $4x^4+x^2+3x+1=0$?
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Ref: R9082

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Question

Prove that, if \(\alpha\) is a repeated root of \(f(x)=0\), where \(f(x)\) is a polynomial, then \(\alpha\) is a root of the equation \(f'(x)=0\).

Given that the equation \(4x^4+x^2+3x+1=0\) has a repeated root, find its value.