A strange sequence of polynomials over $\mathbb{F}_{7}$

Question

Let $P=X^{10} +5X^{5}+1$ and $Q=5X^{8}+6X^{3}$ in $\\mathbb{F}{7}[X]$. How we can prove this strange relation of Euclid division $$U{2i}P-X^{4}U_{2i-1}^{7}=4.3^{2i-1}Q$