Word Count : 559

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We are asked to find the GCD of the polynomials: $$ f(x) = x^4 + 3x + 2 \quad \text{and} \quad g(x) = x^3 + 3x^2 + 5x + 3 $$ We'll use polynomial division (Euclidean algorithm) to find their greatest common divisor (GCD). --- ### Step 1: Divide $f(x) \div g(x)$ We divide $f(x) = x^4 + 0x^3 + 0x^2 + 3x + 2$ by $g(x) = x^3 + 3x^2 + 5x + 3$ Leading terms: $x^4 \div x^3 = x$ Multiply $g(x)$ by $x$: $$ x(x^3 + 3x^2 + 5x + 3) = x^4 + 3x^3 + 5x^2 + 3x $$ Subtract from $f(x)$: $$ (x^4 + 0x^3 + _________ ___ ___ ____ _________ _____ ______ ____ __________.
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