Express the following in the form a+bi
`(2 +3i) (3+7i)`

Answer

To express the product of the complex numbers \( (2 + 3i)(3 + 7i) \) in the form \( a + bi \), we will follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ (2 + 3i)(3 + 7i) \]

### Step 2: Apply the distributive property (FOIL method)
We will multiply each term in the first bracket by each term in the second bracket:
\[ = 2 \cdot 3 + 2 \cdot 7i + 3i \cdot 3 + 3i \cdot 7i \]

### Step 3: Calculate each term
Calculating each term:
- \( 2 \cdot 3 = 6 \)
- \( 2 \cdot 7i = 14i \)
- \( 3i \cdot 3 = 9i \)
- \( 3i \cdot 7i = 21i^2 \)

### Step 4: Combine like terms
Now, we combine the real and imaginary parts:
\[ = 6 + 14i + 9i + 21i^2 \]

### Step 5: Substitute \( i^2 \) with \(-1\)
Since \( i^2 = -1 \), we substitute:
\[ = 6 + 14i + 9i + 21(-1) \]
\[ = 6 + 14i + 9i - 21 \]

### Step 6: Simplify the expression
Now, we combine the real parts and the imaginary parts:
\[ = (6 - 21) + (14i + 9i) \]
\[ = -15 + 23i \]

### Final Result
Thus, the expression \( (2 + 3i)(3 + 7i) \) in the form \( a + bi \) is:
\[ \boxed{-15 + 23i} \]