**Introduction**

In algebra, simplifying expressions and finding equivalent forms is a fundamental skill. Equivalent expressions are those that have the same value for all values of the variables involved. This article will explore how to find an expression equivalent to 10x2y+25x210x^2y + 25x^210x2y+25×2, using factoring techniques to simplify the given expression.

**Understanding the Expression**

The expression given is: 10x2y+25x210x^2y + 25x^210x2y+25×2

To find an equivalent expression, we need to factor out the common terms. Factoring involves finding the greatest common factor (GCF) of the terms in the expression and then rewriting the expression in a simpler form.

**Step-by-Step Factoring Process**

**Step 1: Identify the Common Factor**

First, we identify the greatest common factor (GCF) of the terms in the expression. The terms are 10x2y10x^2y10x2y and 25x225x^225×2.

- The numerical part of the GCF: The GCF of 10 and 25 is 5.
- The variable part of the GCF: Both terms contain x2x^2×2.

Therefore, the GCF of the expression is 5x25x^25×2.

**Step 2: Factor Out the GCF**

Next, we factor out 5x25x^25×2 from each term in the expression: 10x2y+25×2=5×2(2y)+5×2(5)10x^2y + 25x^2 = 5x^2(2y) + 5x^2(5)10x2y+25×2=5×2(2y)+5×2(5)

**Step 3: Simplify the Expression**

After factoring out the GCF, we combine the terms inside the parentheses: 5×2(2y+5)5x^2(2y + 5)5×2(2y+5)

Therefore, the expression equivalent to 10x2y+25x210x^2y + 25x^210x2y+25×2 is: 5×2(2y+5)5x^2(2y + 5)5×2(2y+5)

**Conclusion**

The expression 10x2y+25x210x^2y + 25x^210x2y+25×2 can be simplified by factoring out the greatest common factor. The equivalent expression is: 5×2(2y+5)5x^2(2y + 5)5×2(2y+5)

This process of factoring helps to simplify and understand the structure of algebraic expressions. It is a valuable skill for solving equations and simplifying complex expressions in algebra.