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

Question

UpStudy AI · Accepted Answer

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:**

$$
9a^2
$$
The simplified expression is $$9a^2$$.

ar)

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