Simplify `(5a-2b) (25a^(2)+10ab+4b^(2)_) -(2a+5b) (4a^(2)-10ab+25b^(2)).`

Answer

To simplify the expression \((5a-2b)(25a^2 + 10ab + 4b^2) - (2a + 5b)(4a^2 - 10ab + 25b^2)\), we will use algebraic identities and perform the necessary calculations step by step.

### Step 1: Identify the identities

The first part of the expression \((5a - 2b)(25a^2 + 10ab + 4b^2)\) can be recognized as a product of a binomial and a perfect square trinomial. The trinomial can be rewritten as:
\[ (5a)^2 + 2(5a)(2b) + (2b)^2 = (5a + 2b)^2 \]
Thus, we can express the first part as:
\[ (5a - 2b)(5a + 2b)^2 \]

### Step 2: Simplify the first part

Using the identity \(A^2 - B^2 = (A - B)(A + B)\), we can rewrite:
\[ (5a - 2b)(5a + 2b) = 25a^2 - 4b^2 \]
So, the first part simplifies to:
\[ (5a - 2b)(25a^2 + 10ab + 4b^2) = (5a - 2b)(5a + 2b)(5a + 2b) = (25a^2 - 4b^2)(5a + 2b) \]

### Step 3: Simplify the second part

Now, for the second part \((2a + 5b)(4a^2 - 10ab + 25b^2)\), we can recognize that \(4a^2 - 10ab + 25b^2\) is also a perfect square trinomial:
\[ (2a)^2 - 2(2a)(5b) + (5b)^2 = (2a - 5b)^2 \]
Thus, we can express the second part as:
\[ (2a + 5b)(2a - 5b)^2 \]

### Step 4: Simplify the entire expression

Now we can rewrite the entire expression:
\[ (25a^2 - 4b^2)(5a + 2b) - (2a + 5b)(2a - 5b)^2 \]

### Step 5: Expand both parts

1. For the first part:
\[ (25a^2 - 4b^2)(5a + 2b) = 25a^2 \cdot 5a + 25a^2 \cdot 2b - 4b^2 \cdot 5a - 4b^2 \cdot 2b \]
This expands to:
\[ 125a^3 + 50a^2b - 20ab^2 - 8b^3 \]

2. For the second part:
\[ (2a + 5b)(2a - 5b)^2 = (2a + 5b)(4a^2 - 20ab + 25b^2) \]
This expands to:
\[ 8a^3 - 40a^2b + 50ab^2 + 20b^3 \]

### Step 6: Combine the results

Now we combine both parts:
\[ (125a^3 + 50a^2b - 20ab^2 - 8b^3) - (8a^3 - 40a^2b + 50ab^2 + 20b^3) \]
This simplifies to:
\[ (125a^3 - 8a^3) + (50a^2b + 40a^2b) + (-20ab^2 - 50ab^2) + (-8b^3 - 20b^3) \]
Which results in:
\[ 117a^3 + 90a^2b - 70ab^2 - 28b^3 \]

### Final Result

Thus, the simplified expression is:
\[ 117a^3 + 90a^2b - 70ab^2 - 28b^3 \]