To determine the degree of the polynomial \( 6x^6 + 9x^3 + 3x^2 - 4x^{10} - 9x^5 - 5x^6 \), we first combine like terms and simplify the expression.
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Combine the \( x^6 \) terms: \[ 6x^6 - 5x^6 = 1x^6 \]
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The polynomial now looks like: \[ -4x^{10} + 1x^6 - 9x^5 + 9x^3 + 3x^2 \]
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The degree of a polynomial is defined as the highest power of \( x \) that appears in the polynomial.
In this case, the highest power is \( x^{10} \) from the term \( -4x^{10} \).
Therefore, the degree of the polynomial is \( \boxed{10} \).