Simplify :
`(3y + 4z)(3y - 4z) + (2y + 7z)(y+z)`

Answer

To simplify the expression \((3y + 4z)(3y - 4z) + (2y + 7z)(y + z)\), we will follow these steps:

### Step 1: Apply the difference of squares formula
The first part of the expression, \((3y + 4z)(3y - 4z)\), can be simplified using the identity \(a^2 - b^2\), where \(a = 3y\) and \(b = 4z\).

\[ (3y + 4z)(3y - 4z) = (3y)^2 - (4z)^2 = 9y^2 - 16z^2 \]

### Step 2: Expand the second part of the expression
Now, we will expand the second part of the expression, \((2y + 7z)(y + z)\).

Using the distributive property (also known as the FOIL method for binomials):

\[ (2y + 7z)(y + z) = 2y \cdot y + 2y \cdot z + 7z \cdot y + 7z \cdot z \]
\[ = 2y^2 + 2yz + 7yz + 7z^2 \]
\[ = 2y^2 + (2yz + 7yz) + 7z^2 = 2y^2 + 9yz + 7z^2 \]

### Step 3: Combine the results
Now we will combine the results from Step 1 and Step 2:

\[ 9y^2 - 16z^2 + 2y^2 + 9yz + 7z^2 \]

### Step 4: Combine like terms
Now, we will combine the like terms:

- For \(y^2\): \(9y^2 + 2y^2 = 11y^2\)
- For \(z^2\): \(-16z^2 + 7z^2 = -9z^2\)
- For \(yz\): \(9yz\) remains as it is.

Putting it all together, we get:

\[ 11y^2 + 9yz - 9z^2 \]

### Final Answer:
Thus, the simplified expression is:

\[ \boxed{11y^2 + 9yz - 9z^2} \]