Factorise:
`(2x +5y)^2-1`

Answer

To factorise the expression \((2x + 5y)^2 - 1\), we can use the difference of squares formula. Here’s the step-by-step solution:

### Step 1: Identify the expression as a difference of squares
The expression \((2x + 5y)^2 - 1\) can be recognized as a difference of squares, which follows the formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
In our case, we can identify:
- \(a = (2x + 5y)\)
- \(b = 1\)

### Step 2: Apply the difference of squares formula
Using the difference of squares formula, we can rewrite the expression:
\[ (2x + 5y)^2 - 1^2 = ((2x + 5y) - 1)((2x + 5y) + 1) \]

### Step 3: Simplify the factors
Now we simplify the factors:
1. The first factor:
\[ (2x + 5y) - 1 = 2x + 5y - 1 \]
2. The second factor:
\[ (2x + 5y) + 1 = 2x + 5y + 1 \]

### Step 4: Write the final factored form
Putting it all together, we have:
\[ (2x + 5y - 1)(2x + 5y + 1) \]

Thus, the factored form of the expression \((2x + 5y)^2 - 1\) is:
\[ (2x + 5y - 1)(2x + 5y + 1) \]

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