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 to group similar terms together: \[ x^{2} + 2xy + y^{2} + 3x + 3y - 18 \] This can be rewritten as: \[ (x^{2} + 2xy + y^{2}) + (3x + 3y) - 18 \] ### Step 2: Recognizing a Perfect Square Notice that \(x^{2} + 2xy + y^{2}\) is a perfect square trinomial. It can be factored as: \[ (x + y)^{2} \] So, we can rewrite the expression as: \[ (x + y)^{2} + 3(x + y) - 18 \] ### Step 3: Substituting for Simplicity Let’s make a substitution to simplify our expression. Let: \[ z = x + y \] Then, the expression becomes: \[ z^{2} + 3z - 18 \] ### Step 4: Factoring the Quadratic Now, 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: \[ 6 \times (-3) = -18 \quad \text{and} \quad 6 + (-3) = 3 \] Thus, we can factor the quadratic as: \[ (z + 6)(z - 3) \] ### Step 5: Substituting Back Now, we substitute back \(z = x + y\): \[ (z + 6)(z - 3) = (x + y + 6)(x + y - 3) \] ### Step 6: Final Factorization Thus, the complete factorization of the original expression \(x^{2} + 2xy + y^{2} + 3x + 3y - 18\) 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 align with our derived factors. - **Option D: (x+y-6)(x+y+3)**: This option also does not match our factorization. The signs of the constants are incorrect. ### Revision Summary - Recognize perfect square trinomials and factor them. - Use substitution to simplify complex expressions. - Factor quadratics by finding two numbers that multiply to the constant term and add to the linear coefficient. - Always substitute back to the original variables after factoring.