## Simplifying the Expression: (a + 4)^2 + 2(a + 4) + 1 / (a + 5)

This expression combines several algebraic concepts, including:

**Squaring a binomial:**(a + 4)^2**Distributing:**2(a + 4)**Adding fractions:**1 / (a + 5)

Let's break down each part and simplify the entire expression:

### Simplifying (a + 4)^2

We can use the formula for squaring a binomial: **(x + y)^2 = x^2 + 2xy + y^2**

Applying this to our case:

(a + 4)^2 = a^2 + 2(a)(4) + 4^2 = a^2 + 8a + 16

### Simplifying 2(a + 4)

Using the distributive property:

2(a + 4) = 2a + 8

### Combining the Terms

Now we can combine all the simplified parts:

(a + 4)^2 + 2(a + 4) + 1 / (a + 5) = a^2 + 8a + 16 + 2a + 8 + 1 / (a + 5)

### Simplifying Further

Combine like terms:

a^2 + 10a + 24 + 1 / (a + 5)

To express this as a single fraction, we need to find a common denominator:

(a^2 + 10a + 24)(a + 5) / (a + 5) + 1 / (a + 5)

Now we can combine the numerators:

**(a^2 + 10a + 24)(a + 5) + 1** / (a + 5)

### Final Result

The simplified expression is:

**(a^2 + 10a + 24)(a + 5) + 1** / **(a + 5)**

**Important Note:** This expression is undefined when a = -5, as it would result in division by zero.