To factor the expression \( x^2 + 2xy + y^2 + 3x + 3y - 18 \) completely, we will follow a systematic approach. ### Step 1: Rearranging the Expression First, let's rearrange the expression for clarity: \[ x^2 + 2xy + y^2 + 3x + 3y - 18 \] We can group the first three terms together, which form a perfect square trinomial: \[ (x^2 + 2xy + y^2) + (3x + 3y) - 18 \] This simplifies to: \[ (x+y)^2 + 3(x+y) - 18 \] ### Step 2: Substituting for Simplicity Let \( z = x + y \). Then, we can rewrite the expression as: \[ z^2 + 3z - 18 \] ### Step 3: Factoring the Quadratic Next, we need to factor the quadratic \( z^2 + 3z - 18 \). We are looking for two numbers that multiply to \(-18\) (the constant term) and add to \(3\) (the coefficient of \(z\)). The numbers \(6\) and \(-3\) satisfy these conditions: \[ z^2 + 6z - 3z - 18 = (z + 6)(z - 3) \] ### Step 4: Substituting Back Now, we substitute back \( z = x + y \): \[ (z + 6)(z - 3) = (x + y + 6)(x + y - 3) \] ### Step 5: Final Factorization Thus, the complete factorization of the original expression is: \[ (x + y + 6)(x + y - 3) \] ### Conclusion: Correct Option The correct option is **A: (x+y+6)(x+y-3)**. ### Explanation of Other Options - **Option B: (x-y-6)(x-y+3)**: This option is incorrect because it does not match the structure of our factorization. The signs and terms do not correspond to the original expression. - **Option C: (x-y+6)(x-y-3)**: Similar to option B, this option incorrectly uses \(y\) with a negative sign, which does not reflect the original expression's terms. - **Option D: (x+y-6)(x+y+3)**: This option also does not match our factorization. The signs are incorrect, and it does not yield the original expression when expanded. ### Revision Summary - Factor the expression by grouping and recognizing perfect squares. - Substitute variables to simplify the quadratic. - Factor the quadratic using the product-sum method. - Substitute back to get the final factorization. By following these steps, you can effectively factor similar expressions in the future!