ar) \( (2 a+3 b)^{2}-2(2 a+3 b)(3 b-a)+(3 b-a)^{2} \)

Let's simplify the expression step by step: \[ (2a + 3b)^2 - 2(2a + 3b)(3b - a) + (3b - a)^2 \] ### Step 1: Recognize the Pattern Notice that the expression resembles the expanded form of \((A - B)^2\), where: \[ A = 2a + 3b \quad \text{and} \quad B = 3b - a \] So, the expression becomes: \[ A^2 - 2AB + B^2 = (A - B)^2 \] ### Step 2: Compute \(A - B\) \[ A - B = (2a + 3b) - (3b - a) = 2a + 3b - 3b + a = 3a \] ### Step 3: Square the Result \[ (A - B)^2 = (3a)^2 = 9a^2 \] ### Alternative Method: Expanding Each Term If you prefer expanding each term individually: 1. **Expand \((2a + 3b)^2\):** \[ (2a + 3b)^2 = 4a^2 + 12ab + 9b^2 \] 2. **Expand \(-2(2a + 3b)(3b - a)\):** \[ (2a + 3b)(3b - a) = 6ab - 2a^2 + 9b^2 - 3ab = 3ab - 2a^2 + 9b^2 \] \[ -2(3ab - 2a^2 + 9b^2) = -6ab + 4a^2 - 18b^2 \] 3. **Expand \((3b - a)^2\):** \[ (3b - a)^2 = 9b^2 - 6ab + a^2 \] 4. **Combine all terms:** \[ 4a^2 + 12ab + 9b^2 - 6ab + 4a^2 - 18b^2 + 9b^2 - 6ab + a^2 = 9a^2 \] Both methods lead to the same result. **Final Answer:** \[ 9a^2 \]