Find the expansion of the using suitable identity.
`((4x )/( 5 ) + (y)/(4) ) ( ( 4x )/( 5) + ( 3y)/(4))`

Answer

To find the expansion of the expression \(\left(\frac{4x}{5} + \frac{y}{4}\right) \left(\frac{4x}{5} + \frac{3y}{4}\right)\) using a suitable identity, we can use the identity for the product of two binomials:

\[ (a + b)(c + d) = ac + ad + bc + bd \]

### Step-by-Step Solution:

1. **Identify the terms**:
Let \(a = \frac{4x}{5}\), \(b = \frac{y}{4}\), \(c = \frac{4x}{5}\), and \(d = \frac{3y}{4}\).

2. **Apply the identity**:
Using the identity, we can expand the expression:
\[ \left(\frac{4x}{5} + \frac{y}{4}\right) \left(\frac{4x}{5} + \frac{3y}{4}\right) = ac + ad + bc + bd \]

3. **Calculate each term**:
- **First term**:
\[ ac = \left(\frac{4x}{5}\right) \left(\frac{4x}{5}\right) = \frac{16x^2}{25} \]
- **Second term**:
\[ ad = \left(\frac{4x}{5}\right) \left(\frac{3y}{4}\right) = \frac{12xy}{20} = \frac{3xy}{5} \]
- **Third term**:
\[ bc = \left(\frac{y}{4}\right) \left(\frac{4x}{5}\right) = \frac{4xy}{20} = \frac{xy}{5} \]
- **Fourth term**:
\[ bd = \left(\frac{y}{4}\right) \left(\frac{3y}{4}\right) = \frac{3y^2}{16} \]

4. **Combine all terms**:
Now, combine all the calculated terms:
\[ \frac{16x^2}{25} + \frac{3xy}{5} + \frac{xy}{5} + \frac{3y^2}{16} \]
Combine the \(xy\) terms:
\[ \frac{16x^2}{25} + \left(\frac{3xy}{5} + \frac{xy}{5}\right) + \frac{3y^2}{16} = \frac{16x^2}{25} + \frac{4xy}{5} + \frac{3y^2}{16} \]

5. **Final expression**:
The final expanded expression is:
\[ \frac{16x^2}{25} + \frac{4xy}{5} + \frac{3y^2}{16} \]