To find the area of a rectangle using the expression \( 25a^2 - 35a + 12 \), we first need to understand the context in which this expression represents the area.

Understanding the Quadratic Expression
The expression \( 25a^2 - 35a + 12 \) is a quadratic polynomial. In the context of geometry, particularly when discussing rectangles, this polynomial might represent the area in terms of its length and width.

Factoring the Expression
To explore the dimensions further, we can factor the quadratic expression:
- **Step 1:** Identify coefficients:
- \( A = 25 \)
- \( B = -35 \)
- \( C = 12 \)
- **Step 2:** Look for two numbers that multiply to \( A \times C = 25 \times 12 = 300 \) and add to \( B = -35 \).
- **Step 3:** These numbers are \( -15 \) and \( -20 \).
- **Step 4:** Rewrite the expression:
- \( 25a^2 - 15a - 20a + 12 \)
- **Step 5:** Factor by grouping:
- \( 5a(5a - 3) - 4(5a - 3) \)
- \( (5a - 4)(5a - 3) \)

Dimensions of the Rectangle
The factors \( (5a - 4) \) and \( (5a - 3) \) represent the length and width of the rectangle.

Calculating the Area
Thus, the area \( A \) of the rectangle is given by:
- **Area Formula:**
- \( A = \text{Length} \times \text{Width} = (5a - 4)(5a - 3) \)
In conclusion, the area of the rectangle represented by the expression \( 25a^2 - 35a + 12 \) is \( (5a - 4)(5a - 3) \) square units.