`25a^(2)-4b^(2)+5a+2b=` ______

Options

A. `(5a+2b-1)(5a-2b)`
B. `(5a+2b+1)(5a-2b)`
C. `(5a+2b)(5a-2b-1)`
D. `(5a+2b)(5a-2b+1)`

Answer

To solve the expression \( 25a^2 - 4b^2 + 5a + 2b \), we will follow these steps:

### Step 1: Rearranging the Expression
We start with the expression:
\[ 25a^2 - 4b^2 + 5a + 2b \]

### Step 2: Grouping Terms
We can rearrange the expression to group the quadratic terms and the linear terms:
\[ (25a^2 - 4b^2) + (5a + 2b) \]

### Step 3: Recognizing a Difference of Squares
The first part \( 25a^2 - 4b^2 \) is a difference of squares. We can rewrite it as:
\[ (5a)^2 - (2b)^2 \]

### Step 4: Applying the Difference of Squares Formula
Using the difference of squares formula \( x^2 - y^2 = (x + y)(x - y) \), we can factor \( (5a)^2 - (2b)^2 \):
\[ (5a + 2b)(5a - 2b) \]

### Step 5: Combining with the Linear Terms
Now, we can combine this with the linear terms \( 5a + 2b \):
\[ (5a + 2b)(5a - 2b) + (5a + 2b) \]

### Step 6: Factoring Out the Common Term
Notice that \( (5a + 2b) \) is a common factor. We can factor it out:
\[ (5a + 2b)((5a - 2b) + 1) \]

### Step 7: Final Expression
Thus, the final factored form of the expression is:
\[ (5a + 2b)(5a - 2b + 1) \]

### Summary
The expression \( 25a^2 - 4b^2 + 5a + 2b \) can be factored as:
\[ (5a + 2b)(5a - 2b + 1) \]

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