Multiply `((1)/(5) x - (1)/(4)y)` and `(5x^(2) - 4y^(2))`

Answer

To solve the problem of multiplying the expressions \(\left(\frac{1}{5}x - \frac{1}{4}y\right)\) and \((5x^2 - 4y^2)\), we will follow these steps:

### Step 1: Distribute the first expression over the second
We will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first expression by each term in the second expression.

\[ \left(\frac{1}{5}x - \frac{1}{4}y\right)(5x^2 - 4y^2) = \frac{1}{5}x \cdot 5x^2 - \frac{1}{5}x \cdot 4y^2 - \frac{1}{4}y \cdot 5x^2 + \frac{1}{4}y \cdot 4y^2 \]

### Step 2: Multiply the terms
Now, we will perform the multiplication for each term:

1. \(\frac{1}{5}x \cdot 5x^2\):
\[ = \frac{5}{5}x^{1+2} = 1x^3 = x^3 \]

2. \(-\frac{1}{5}x \cdot 4y^2\):
\[ = -\frac{4}{5}xy^2 \]

3. \(-\frac{1}{4}y \cdot 5x^2\):
\[ = -\frac{5}{4}x^2y \]

4. \(\frac{1}{4}y \cdot 4y^2\):
\[ = \frac{4}{4}y^{1+2} = 1y^3 = y^3 \]

### Step 3: Combine all the terms
Now, we will combine all the results from the multiplications:

\[ x^3 - \frac{4}{5}xy^2 - \frac{5}{4}x^2y + y^3 \]

### Final Answer
Thus, the final result of multiplying the two expressions is:

\[ x^3 - \frac{4}{5}xy^2 - \frac{5}{4}x^2y + y^3 \]

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