A good use of the remainder theorem

Algebra Level 4

A polynomial \(P(x)\) divided by \( x ^{2}+x+1 \) give us a remainder of \( -x+1\) and when divided by \(x ^ {2} - x +1 \) has a remainder of \( 3x+5 \). Knowing that the remainder of the division of \(P(x)\) for \( x ^ {4} + x ^{2} + 1\) is \( R(x) \) a polynomial of the third degree, evaluate the sum of the squares of the coefficients of \( R(x) \).

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