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To solve the problem, we need to determine the value of \( h \) such that the vector \( \mathbf{b} \) is in the span of the vectors \( \mathbf{u} \) and \( \mathbf{v} \). 

### Given:
\[
\mathbf{b} = \begin{bmatrix} h \\ -3 \\ 2h - 5 \end{bmatrix}, \quad \mathbf{u} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}
\]

### Problem:
The vector \( \mathbf{b} \) is in the span of \( \mathbf{u} \) and \( \mathbf{v} \) if there exist scalars \( a \) and \( b \) such that:
\[
\mathbf{b} = a\mathbf{u} + b\mathbf{v}
\]
This expands to:
\[
\begin{bmatrix} h \\ -3 \\ 2h - 5 \end{bmatrix} = a\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} + b\begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}
\]

### Expanding the equation:
\[
\begin{bmatrix} h \\ -3 \\ 2h - 5 \end{bmatrix} = \begin{bmatrix} a + 2b \\ b \\ 3b \end{bmatrix}
\]

### This gives us the system of equations:
1. \( h = a + 2b \)
2. \( -3 = b \)
3. \( 2h - 5 = 3b \)

### Solve the system:

1. From equation \( 2 \), we have \( b = -3 \).
2. Substitute \( b = -3 \) into the third equation:
   \[
   2h - 5 = 3(-3) = -9
   \]
   \[
   2h = -9 + 5 = -4
   \]
   \[
   h = -2
   \]

3. Finally, substitute \( h = -2 \) and \( b = -3 \) into the first equation:
   \[
   -2 = a + 2(-3)
   \]
   \[
   -2 = a - 6
   \]
   \[
   a = 4
   \]

### Conclusion:
The value of \( h \) that ensures \( \mathbf{b} \) is in the span of \( \mathbf{u} \) and \( \mathbf{v} \) is \( \boxed{-2} \).

This corresponds to option **d)**.

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Would you like more details or have any further questions? Here are some related questions you might find useful:

1. What does it mean for a vector to be in the span of other vectors?
2. How can you determine if a set of vectors are linearly independent?
3. What is the geometric interpretation of the span of two vectors?
4. Can you have more than one solution for the scalars in a span problem?
5. How does the dimension of the vector space affect the span?

**Tip:** Always check if a vector can be expressed as a linear combination of other vectors by setting up a system of linear equations.

Let b = [h; -3; 2h - 5], u = [1; 0; 0], and v = [2; 1; 3]. For what value of h is b in the span of u and v?

This problem demonstrates how to find the value of h such that the vector b = [h; -3; 2h - 5] is in the span of two vectors u and v. Using concepts from linear algebra, including vector spaces, spans, and solving systems of equations, this step-by-step solution shows how h = -2.

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