Find the product. `(3x-5y-4)(9x^2+25y^2+15xy-20y+12x+16)`

Answer

To find the product of the expressions \((3x - 5y - 4)(9x^2 + 25y^2 + 15xy - 20y + 12x + 16)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write the expressions clearly
We start with the expression:
\[ (3x - 5y - 4)(9x^2 + 25y^2 + 15xy - 20y + 12x + 16) \]

### Step 2: Distribute each term in the first polynomial to each term in the second polynomial
We will distribute \(3x\), \(-5y\), and \(-4\) to every term in the second polynomial.

1. **Distributing \(3x\)**:
\[ 3x \cdot 9x^2 = 27x^3 \]
\[ 3x \cdot 25y^2 = 75xy^2 \]
\[ 3x \cdot 15xy = 45x^2y \]
\[ 3x \cdot (-20y) = -60xy \]
\[ 3x \cdot 12x = 36x^2 \]
\[ 3x \cdot 16 = 48x \]

2. **Distributing \(-5y\)**:
\[ -5y \cdot 9x^2 = -45yx^2 \]
\[ -5y \cdot 25y^2 = -125y^3 \]
\[ -5y \cdot 15xy = -75y^2x \]
\[ -5y \cdot (-20y) = 100y^2 \]
\[ -5y \cdot 12x = -60xy \]
\[ -5y \cdot 16 = -80y \]

3. **Distributing \(-4\)**:
\[ -4 \cdot 9x^2 = -36x^2 \]
\[ -4 \cdot 25y^2 = -100y^2 \]
\[ -4 \cdot 15xy = -60xy \]
\[ -4 \cdot (-20y) = 80y \]
\[ -4 \cdot 12x = -48x \]
\[ -4 \cdot 16 = -64 \]

### Step 3: Combine all the terms
Now, we will combine all the terms we obtained:
\[ 27x^3 + (75xy^2 + 45x^2y - 45yx^2 - 125y^3 - 75y^2x + 100y^2 - 36x^2 - 60xy - 80y - 48x - 64) \]

### Step 4: Group like terms
Combining like terms:
- \(x^3\) terms: \(27x^3\)
- \(y^3\) terms: \(-125y^3\)
- \(xy^2\) terms: \(75xy^2\)
- \(x^2y\) terms: \(45x^2y - 45yx^2 - 75y^2x = 0\) (they cancel out)
- \(x^2\) terms: \(-36x^2\)
- \(xy\) terms: \(-60xy - 60xy = -120xy\)
- \(y^2\) terms: \(100y^2\)
- \(x\) terms: \(48x - 48x = 0\) (they cancel out)
- Constant term: \(-64\)

### Final Expression
Putting it all together, we get:
\[ 27x^3 - 125y^3 - 120xy + 100y^2 - 64 \]

### Final Answer
The product of the expressions is:
\[ 27x^3 - 125y^3 - 120xy + 100y^2 - 64 \]