1. **Problem Statement:** Factorize the expression $$m^2 - 6m - 18mk + 108k$$. 2. **Identify the terms:** The expression has four terms: $$m^2$$, $$-6m$$, $$-18mk$$, and $$108k$$. 3. **Group terms:** Group the terms to factor by grouping: $$ (m^2 - 6m) - (18mk - 108k) $$ 4. **Factor each group:** - From $$m^2 - 6m$$, factor out $$m$$: $$ m(m - 6) $$ - From $$-18mk + 108k$$, factor out $$-18k$$: $$ -18k(m - 6) $$ 5. **Factor out the common binomial:** Both groups contain $$ (m - 6) $$, so factor it out: $$ (m - 6)(m - 18k) $$ 6. **Check the factorization:** Multiply back to verify: $$ (m - 6)(m - 18k) = m^2 - 18mk - 6m + 108k $$ This matches the original expression except for the sign of the $$18mk$$ term. 7. **Correction:** The original expression has $$-18mk$$, but the factorization given is $$ (m - 6)(m + 18k) $$. Multiplying $$ (m - 6)(m + 18k) $$ gives: $$ m^2 + 18mk - 6m - 108k $$ which does not match the original expression. **Therefore, the correct factorization is:** $$ (m - 6)(m - 18k) $$ **Summary:** The expression factorizes to $$ (m - 6)(m - 18k) $$, not $$ (m - 6)(m + 18k) $$.