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To solve the expression \((2x + 5y)(3x - 2y)(5x + y)\), we need to expand and fully simplify it step by step:

### Step 1: Expand the First Two Binomials
First, we'll expand \((2x + 5y)(3x - 2y)\).

\[
(2x + 5y)(3x - 2y) = (2x)(3x) + (2x)(-2y) + (5y)(3x) + (5y)(-2y)
\]

\[
= 6x^2 - 4xy + 15xy - 10y^2
\]

Combine like terms:

\[
= 6x^2 + 11xy - 10y^2
\]

### Step 2: Multiply the Result by the Third Binomial
Now, multiply \((6x^2 + 11xy - 10y^2)\) by \((5x + y)\).

\[
(6x^2 + 11xy - 10y^2)(5x + y) 
\]

Distribute each term in \((6x^2 + 11xy - 10y^2)\) with both terms in \((5x + y)\).

#### Expand Each Term
1. **First Term: \(6x^2 \cdot (5x + y)\)**:
   \[
   = 6x^2 \cdot 5x + 6x^2 \cdot y = 30x^3 + 6x^2y
   \]

2. **Second Term: \(11xy \cdot (5x + y)\)**:
   \[
   = 11xy \cdot 5x + 11xy \cdot y = 55x^2y + 11xy^2
   \]

3. **Third Term: \(-10y^2 \cdot (5x + y)\)**:
   \[
   = -10y^2 \cdot 5x + (-10y^2) \cdot y = -50xy^2 - 10y^3
   \]

### Step 3: Combine All Terms
Collect and combine all the terms obtained:

\[
= 30x^3 + (6x^2y + 55x^2y) + (11xy^2 - 50xy^2) - 10y^3
\]

\[
= 30x^3 + 61x^2y - 39xy^2 - 10y^3
\]

### Final Answer
The fully expanded and simplified form is:

\[
\boxed{30x^3 + 61x^2y - 39xy^2 - 10y^3}
\]

Would you like any further explanation or details on any part of the solution?

Here are some related questions you might find interesting:
1. How would the result change if one of the signs in the binomials was altered?
2. What happens if you expand using a different sequence, like the third binomial first?
3. Can the expression be factored back into its binomial components?
4. How would you approach expanding if more variables were involved?
5. What techniques can help quickly identify terms to combine?

**Tip:** Always combine like terms carefully to ensure accuracy when expanding and simplifying polynomial expressions.

Expand and fully simplify (2x + 5y)(3x - 2y)(5x + y).

Learn how to expand and fully simplify the expression (2x + 5y)(3x - 2y)(5x + y). This algebra problem involves polynomial multiplication and combining like terms, suitable for students in Grades 9-12.

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