Let's simplify the given expressions:

a. $\frac{4 + i}{7 + i}$

To simplify this complex fraction, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $7 + i$ is $7 - i$.

$\frac{4 + i}{7 + i} \times \frac{7 - i}{7 - i} = \frac{(4 + i)(7 - i)}{(7 + i)(7 - i)}$

First, simplify the denominator:

$(7 + i)(7 - i) = 7^2 - i^2 = 49 - (-1) = 50$

Now, expand the numerator:

$(4 + i)(7 - i) = 4 \times 7 - 4 \times i + i \times 7 - i \times i = 28 - 4i + 7i - i^2$

Simplify the expression:

$= 28 + 3i + 1 = 29 + 3i$

So, the simplified form is:

$\frac{4 + i}{7 + i} = \frac{29 + 3i}{50} = \frac{29}{50} + \frac{3i}{50}$

b. $(12 + 6i) + (9 + 5i)$

To simplify this expression, just add the real parts and the imaginary parts separately:

$(12 + 6i) + (9 + 5i) = (12 + 9) + (6i + 5i) = 21 + 11i$

So the simplified expression is:

$21 + 11i$

Would you like more details or have any questions? Here are some related problems for practice:

1. Simplify $\frac{3 + 2i}{4 - 3i}$.
2. Add $(8 + 2i) + (5 - 3i)$.
3. Simplify $\frac{5 - 4i}{2 + 3i}$.
4. Subtract $(7 + 9i) - (3 + 2i)$.
5. Multiply $(2 + 3i)(4 - i)$.

Tip: When dealing with complex numbers, always remember to simplify the denominator by multiplying by the conjugate to eliminate the imaginary part in the denominator.