1. **State the problem:** Factor out the greatest common factor (GCF) from the polynomial $$5x^4 - 25x^3 + 10x^2$$. 2. **Identify the GCF:** Look at the coefficients 5, 25, and 10. The greatest common factor of these numbers is 5. 3. **Identify the variable part of the GCF:** The terms have powers of $x$ as $x^4$, $x^3$, and $x^2$. The smallest power of $x$ is $x^2$, so the variable part of the GCF is $x^2$. 4. **Write the GCF:** The GCF is $$5x^2$$. 5. **Factor out the GCF:** Divide each term by $5x^2$: $$5x^4 \div 5x^2 = \cancel{5}x^{4-2} = x^2$$ $$-25x^3 \div 5x^2 = -\cancel{5}5x^{3-2} = -5x$$ $$10x^2 \div 5x^2 = \cancel{5}2\cancel{x^2} = 2$$ 6. **Write the factored form:** $$5x^4 - 25x^3 + 10x^2 = 5x^2(x^2 - 5x + 2)$$ **Final answer:** $$5x^2(x^2 - 5x + 2)$$