## Factoring a Perfect Square Trinomial

The expression **(9x - 1)² + (1 - 5x)² + 2(9x - 1)(1 - 5x)** appears to be complex, but it can be simplified by recognizing its pattern. This expression is a perfect square trinomial, which is a special type of trinomial that results from squaring a binomial.

Here's how to identify and factor this expression:

**1. Recognizing the Pattern:**

The pattern of a perfect square trinomial is:

**(a + b)² = a² + 2ab + b²** or **(a - b)² = a² - 2ab + b²**

In our expression, we have:

**(9x - 1)²**: This is the square of the first term (a²).**(1 - 5x)²**: This is the square of the second term (b²).**2(9x - 1)(1 - 5x)**: This is twice the product of the first and second terms (2ab).

**2. Applying the Pattern:**

Let's substitute **a = (9x - 1)** and **b = (1 - 5x)** into the perfect square trinomial pattern:

**(a + b)² = a² + 2ab + b²**

**(9x - 1 + 1 - 5x)² = (9x - 1)² + 2(9x - 1)(1 - 5x) + (1 - 5x)²**

**3. Simplifying the Expression:**

Now, we can simplify the expression:

**(4x)² = (9x - 1)² + 2(9x - 1)(1 - 5x) + (1 - 5x)²**

**(4x)² = **(9x - 1)² + (1 - 5x)² + 2(9x - 1)(1 - 5x)**

**Therefore, the factored form of the expression (9x - 1)² + (1 - 5x)² + 2(9x - 1)(1 - 5x) is (4x)².**

**Important Note:** Recognizing patterns like this can significantly simplify algebraic expressions and make solving equations easier.