Expand an evaivate (1) \( (3 a+b)+2 a(3 a-b) \) \( 2\left[(x+y)^{2}+x y(x+y)\right. \) \( 3\left[(3 x+y)^{2}-(3 x-y)^{2}\right]^{2} \)

Sure, let's expand and evaluate each of the given algebraic expressions step by step. --- ### **1. Expand and Simplify \( (3a + b) + 2a(3a - b) \)** **Step 1: Distribute the \( 2a \) across \( (3a - b) \)** \[ 2a \cdot 3a = 6a^2 \] \[ 2a \cdot (-b) = -2ab \] **Step 2: Combine all terms** \[ (3a + b) + 6a^2 - 2ab = 6a^2 + 3a + b - 2ab \] **Final Expanded Form:** \[ 6a^2 + 3a - 2ab + b \] or, rearranged, \[ 6a^2 + 3a - 2ab + b \] --- ### **2. Expand and Simplify \( 2\left[(x + y)^2 + xy(x + y)\right] \)** **Step 1: Expand \( (x + y)^2 \)** \[ (x + y)^2 = x^2 + 2xy + y^2 \] **Step 2: Expand \( xy(x + y) \)** \[ xy(x + y) = x^2y + xy^2 \] **Step 3: Combine the expanded terms inside the brackets** \[ (x^2 + 2xy + y^2) + (x^2y + xy^2) = x^2 + 2xy + y^2 + x^2y + xy^2 \] **Step 4: Multiply each term by 2** \[ 2(x^2) + 2(2xy) + 2(y^2) + 2(x^2y) + 2(xy^2) = 2x^2 + 4xy + 2y^2 + 2x^2y + 2xy^2 \] **Final Expanded Form:** \[ 2x^2 + 4xy + 2y^2 + 2x^2y + 2xy^2 \] --- ### **3. Expand and Simplify \( 3\left[(3x + y)^2 - (3x - y)^2\right]^2 \)** **Step 1: Expand \( (3x + y)^2 \) and \( (3x - y)^2 \)** \[ (3x + y)^2 = 9x^2 + 6xy + y^2 \] \[ (3x - y)^2 = 9x^2 - 6xy + y^2 \] **Step 2: Subtract the two expanded expressions** \[ (9x^2 + 6xy + y^2) - (9x^2 - 6xy + y^2) = 12xy \] **Step 3: Square the result** \[ (12xy)^2 = 144x^2y^2 \] **Step 4: Multiply by 3** \[ 3 \cdot 144x^2y^2 = 432x^2y^2 \] **Final Expanded Form:** \[ 432x^2y^2 \] --- ### **Summary of Expanded Expressions** 1. **\( (3a + b) + 2a(3a - b) \)** \[ 6a^2 + 3a - 2ab + b \] 2. **\( 2\left[(x + y)^2 + xy(x + y)\right] \)** \[ 2x^2 + 4xy + 2y^2 + 2x^2y + 2xy^2 \] 3. **\( 3\left[(3x + y)^2 - (3x - y)^2\right]^2 \)** \[ 432x^2y^2 \] If you have specific values for the variables and need to evaluate these expressions numerically, please provide them, and I'd be happy to help further!