Let's break down the problem step by step for each part.

The function given is:

$h(x) = \frac{2x + 1}{x + 5}$

(a) Finding $h(11)$:

We need to substitute $x = 11$ into the function:

$h(11) = \frac{2(11) + 1}{11 + 5} = \frac{22 + 1}{16} = \frac{23}{16}$

So, the answer to (a) is:

$\boxed{\frac{23}{16}}$

(b) Finding $3 + 2h(11)$:

From part (a), we know that $h(11) = \frac{23}{16}$. Now, substitute that into the expression:

$3 + 2h(11) = 3 + 2 \times \frac{23}{16} = 3 + \frac{46}{16}$

Convert 3 to a fraction with a denominator of 16:

$3 = \frac{48}{16}$

Now add:

$3 + 2h(11) = \frac{48}{16} + \frac{46}{16} = \frac{94}{16}$

Simplify the fraction:

$\frac{94}{16} = \frac{47}{8}$

So, the answer to (b) is:

$\boxed{\frac{47}{8}}$

(c) Excluding values from the domain of $h(x)$:

The function $h(x)$ is undefined when the denominator is zero. The denominator is $x + 5$, so we set it equal to zero to find the excluded value:

$x + 5 = 0 \quad \Rightarrow \quad x = -5$

So, the value of $x$ that must be excluded from the domain is:

$\boxed{-5}$

(d) Solving $h(x) = 1$:

We need to solve the equation $h(x) = 1$, meaning:

$\frac{2x + 1}{x + 5} = 1$

Multiply both sides by $x + 5$ to eliminate the denominator:

$2x + 1 = (x + 5)$

Simplify the equation:

$2x + 1 = x + 5$

Subtract $x$ from both sides:

$x + 1 = 5$

Now subtract 1 from both sides:

$x = 4$

So, the solution to $h(x) = 1$ is:

$\boxed{4}$

Summary of Answers:

(a) $h(11) = \frac{23}{16}$

(b) $3 + 2h(11) = \frac{47}{8}$

(c) The value $x = -5$ must be excluded from the domain.

(d) The solution to $h(x) = 1$ is $x = 4$.

Would you like more details on any part? Here are 5 related questions to explore further:

1. How can we determine the range of the function $h(x)$?
2. What is the behavior of the function $h(x)$ as $x \to -5$?
3. Can you find the inverse of the function $h(x)$?
4. What is the derivative of the function $h(x)$?
5. What are the critical points of the function $h(x)$?

Tip: Always check for values that make the denominator zero when dealing with rational functions.