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).

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.