To factor the expression \( 4abx - 2axy - 12b^2x + 6bxy \) completely, we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Grouping Terms First, we can group the terms in pairs to make factoring easier: \[ (4abx - 2axy) + (-12b^2x + 6bxy) \] ### Step 2: Factor Out Common Factors Now, we will factor out the common factors from each group. 1. From the first group \( 4abx - 2axy \): - The common factor is \( 2ax \). - Factoring it out gives us: \[ 2ax(2b - y) \] 2. From the second group \( -12b^2x + 6bxy \): - The common factor is \( -6bx \). - Factoring it out gives us: \[ -6bx(2b - y) \] Now, we can rewrite the expression as: \[ 2ax(2b - y) - 6bx(2b - y) \] ### Step 3: Factor Out the Common Binomial Notice that both terms now have a common factor of \( (2b - y) \): \[ (2b - y)(2ax - 6bx) \] ### Step 4: Factor the Remaining Expression Next, we need to factor the expression \( 2ax - 6bx \): - The common factor here is \( 2x \): \[ 2x(a - 3b) \] ### Step 5: Combine Everything Putting it all together, we have: \[ (2b - y)(2x(a - 3b)) \] ### Final Factorization Thus, the complete factorization of the original expression is: \[ 2x(a - 3b)(2b - y) \] ### Conclusion The correct option is **A: \( 2x(a - 3b)(2b - y) \)**. ### Explanation of Other Options - **Option B: \( 2x(3b - a)(2b - y) \)**: This is incorrect because the factor \( (3b - a) \) is not equivalent to \( (a - 3b) \). The signs are reversed, which changes the expression. - **Option C: \( 2x(a - 3b)(y - 2b) \)**: This is incorrect because the factor \( (y - 2b) \) is not the same as \( (2b - y) \). The order of subtraction matters, and this would yield a different expression. - **Option D: \( 2x(2b - a)(3b - y) \)**: This is incorrect because neither \( (2b - a) \) nor \( (3b - y) \) are factors of the original expression. The terms do not match the structure we derived. ### Revision Summary - Factor by grouping terms to identify common factors. - Always check for common binomials after initial factoring. - Be cautious with the order of terms when factoring expressions. - Verify your final factorization by expanding to ensure it matches the original expression.