1. **State the problem:** We want to factor the expression $$(x-7)^7 + 7x(x-7)^6$$ into the form $$(x-7)^6((x-7) + 7)$$. 2. **Identify common factors:** Notice that both terms contain a factor of $$(x-7)^6$$. 3. **Factor out the common term:** $$ (x-7)^7 + 7x(x-7)^6 = (x-7)^6 \cdot \cancel{(x-7)} + 7x (x-7)^6 = (x-7)^6 \left( \cancel{(x-7)} + 7x \right) $$ 4. **Simplify inside the parentheses:** $$ (x-7)^6 \left( (x-7) + 7x \right) = (x-7)^6 \left( x - 7 + 7x \right) = (x-7)^6 (8x - 7) $$ 5. **Compare with the target expression:** The expression you gave is $$(x-7)^6((x-7) + 7)$$, but after factoring and simplifying, the correct factorization is $$(x-7)^6(8x - 7)$$. **Summary:** The step from $$(x-7)^7 + 7x(x-7)^6$$ to $$(x-7)^6((x-7) + 7)$$ is factoring out $$(x-7)^6$$, but the expression inside the parentheses simplifies to $$(8x - 7)$$, not $$(x-7) + 7$$. **Final answer:** $$ (x-7)^7 + 7x(x-7)^6 = (x-7)^6(8x - 7) $$