Q. Which expression is equivalent to \(10x^2y + 25x^2\)? Choices: \(5x^2(2y + 5)\), \(5x^2y(5 + 20y)\), \(10xy(x + 15y)\), \(10x^2(y + 25)\).

Problem: Factor and find which choice is equivalent to the expression

\(10x^2y + 25x^2\)

1. Step 1 — Identify the greatest common numerical factor.

   Compare the coefficients 10 and 25. The greatest common divisor is 5, because 10 = 5·2 and 25 = 5·5.

2. Step 2 — Identify the common variable factors.

   The first term is \(10x^2y\) and the second term is \(25x^2\). Both terms contain \(x^2\). The variable \(y\) appears only in the first term, so it is not a common factor. Thus the common variable factor is \(x^2\).

3. Step 3 — Combine the common numerical and variable factors.

   The common factor is \(5x^2\).

4. Step 4 — Factor out the common factor.

   Write the expression as

   \[
   10x^2y + 25x^2 = 5x^2\!\left(\frac{10x^2y}{5x^2} + \frac{25x^2}{5x^2}\right).
   \]

   Simplify inside the parentheses:

   \[
   \frac{10x^2y}{5x^2} = 2y,\qquad \frac{25x^2}{5x^2} = 5.
   \]

   So the factored form is

   \[
   5x^2(2y + 5).
   \]

5. Step 5 — Verify by expanding (optional check).

   Multiply back: \(5x^2(2y + 5) = 5x^2\cdot 2y + 5x^2\cdot 5 = 10x^2y + 25x^2\). This matches the original expression.

6. Conclusion — Match to the given choices.

   The equivalent expression is

   \(5x^2(2y + 5)\), which corresponds to the first choice written as 5×2(2y + 5).