To factor the expression \( ac - 2bc - a^2 + 4b^2 \) completely, we will follow a systematic approach. ### Step 1: Rearranging the Expression First, let's rearrange the terms for clarity: \[ -ac + 4b^2 - 2bc - a^2 \] This can be rewritten as: \[ -ac - a^2 - 2bc + 4b^2 \] ### Step 2: Grouping Terms Next, we can group the terms in pairs: \[ (-a^2 + ac) + (-2bc + 4b^2) \] Now, we can factor out common factors from each group. ### Step 3: Factoring Each Group 1. From the first group \(-a^2 + ac\), we can factor out \(-a\): \[ -a(a - c) \] 2. From the second group \(-2bc + 4b^2\), we can factor out \(-2b\): \[ -2b(c - 2b) \] Now, we can rewrite the expression as: \[ -a(a - c) - 2b(c - 2b) \] ### Step 4: Combining the Factored Terms Now, we notice that both terms have a common factor of \((a - 2b)\). To see this, we can rewrite the expression: \[ -a(a - c) + 2b(2b - c) \] This can be rearranged to: \[ -(a - 2b)(a - c + 2b) \] ### Step 5: Final Factoring Now, we can factor out \((a - 2b)\): \[ (a - 2b)(c - a + 2b) \] ### Conclusion Thus, the completely factored form of the expression \( ac - 2bc - a^2 + 4b^2 \) is: \[ (a - 2b)(c - a + 2b) \] ### Answer The correct option is **C**: \( (a - 2b)(c - a + 2b) \). ### Explanation of Other Options - **Option A**: \( (a - 2b)(c - a - 2b) \) is incorrect because the second factor does not match the derived factor. - **Option B**: \( (a - 2b)(c + a + 2b) \) is incorrect for the same reason; the second factor is not derived from the original expression. - **Option D**: \( (a - 2b)(c + a - 2b) \) is also incorrect as the second factor does not correspond to the correct factorization. ### Revision Summary - Factor by grouping terms and identifying common factors. - Rearrange the expression for clarity before factoring. - Ensure that all terms are accounted for in the final factorization. - Verify the correctness of the factors by expanding them back to the original expression.