Expand: `(4a- 5b- 2c)^(2)`

Answer

To expand the expression \((4a - 5b - 2c)^2\), we can use the formula for the square of a trinomial, which is given by:

\[ (A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \]

In our case, we can rewrite the expression as:

\[ (4a - 5b - 2c)^2 = (4a + (-5b) + (-2c))^2 \]

Here, we identify:
- \(A = 4a\)
- \(B = -5b\)
- \(C = -2c\)

Now, we can apply the formula step by step.

### Step 1: Calculate \(A^2\), \(B^2\), and \(C^2\)
1. \(A^2 = (4a)^2 = 16a^2\)
2. \(B^2 = (-5b)^2 = 25b^2\)
3. \(C^2 = (-2c)^2 = 4c^2\)

### Step 2: Calculate \(2AB\), \(2BC\), and \(2CA\)
1. \(2AB = 2 \cdot (4a) \cdot (-5b) = -40ab\)
2. \(2BC = 2 \cdot (-5b) \cdot (-2c) = 20bc\)
3. \(2CA = 2 \cdot (-2c) \cdot (4a) = -16ac\)

### Step 3: Combine all the results
Now, we combine all these results together:

\[ (4a - 5b - 2c)^2 = A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \]

Substituting the values we calculated:

\[ = 16a^2 + 25b^2 + 4c^2 - 40ab + 20bc - 16ac \]

### Final Result
Thus, the expanded form of \((4a - 5b - 2c)^2\) is:

\[ 16a^2 + 25b^2 + 4c^2 - 40ab + 20bc - 16ac \]

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