1. **State the problem:** Factor the expression $$Dx^6 + Dx^3 r x^5 + 15 r x^{10}$$. 2. **Rewrite the expression:** Combine like terms and powers of $x$: $$Dx^6 + D r x^{8} + 15 r x^{10}$$ 3. **Identify common factors:** Look for the greatest common factor (GCF) in all terms. - Coefficients: No common factor between $D$, $D r$, and $15 r$ except possibly variables. - Variables: Each term has at least $x^6$. 4. **Factor out the GCF:** Factor out $x^6$: $$x^6 (D + D r x^{2} + 15 r x^{4})$$ 5. **Check inside the parentheses:** The expression inside is: $$D + D r x^{2} + 15 r x^{4}$$ 6. **Look for further factoring:** Group terms: $$D + D r x^{2} + 15 r x^{4} = D(1 + r x^{2}) + 15 r x^{4}$$ No obvious common factor or special product. 7. **Final factored form:** $$x^6 (D + D r x^{2} + 15 r x^{4})$$ This is the fully factored form given the terms.