Multiply matrix B by matrix C and find the distance between points A and B.

Here are the solutions for the given problems:

: Matrix Multiplication

Step 1: Identify the given matrices.
The matrices are given as:

$$B = \begin{pmatrix} -2 & 4 \end{pmatrix}$$ $$C = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$$

Step 2: Determine the dimensions of the matrices.
Matrix B has 1 row and 2 columns, so its dimension is $1 \times 2$.
Matrix C has 2 rows and 1 column, so its dimension is $2 \times 1$.

Step 3: Check if matrix multiplication BC is possible.
For matrix multiplication BC to be defined, the number of columns in B must be equal to the number of rows in C.
Number of columns in B = 2.
Number of rows in C = 2.
Since $2 = 2$, the multiplication is possible, and the resulting matrix BC will have dimensions $1 \times 1$.

Step 4: Perform the matrix multiplication.
To find the product BC, we multiply the elements of the row of B by the corresponding elements of the column of C and sum them.

$$BC = \begin{pmatrix} (-2)(3) + (4)(4) \end{pmatrix}$$ $$BC = \begin{pmatrix} -6 + 16 \end{pmatrix}$$ $$BC = \begin{pmatrix} 10 \end{pmatrix}$$

The final answer is $\boxed{\begin{pmatrix} 10 \end{pmatrix}}$.

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: Distance between two points

Step 1: Identify the given points.
The two points are $A = (-12, 2)$ and $B = (-10, 0)$.
Let $(x_1, y_1) = (-12, 2)$ and $(x_2, y_2) = (-10, 0)$.

Step 2: Recall the distance formula between two points.
The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Step 3: Substitute the coordinates of points A and B into the distance formula.

$$d = \sqrt{(-10 - (-12))^2 + (0 - 2)^2}$$

Step 4: Simplify the terms inside the parentheses.

$$d = \sqrt{(-10 + 12)^2 + (-2)^2}$$ $$d = \sqrt{(2)^2 + (-2)^2}$$

Step 5: Calculate the squares.

$$d = \sqrt{4 + 4}$$

Step 6: Add the terms under the square root.

$$d = \sqrt{8}$$

Step 7: Simplify the square root.

$$d = \sqrt{4 \times 2}$$ $$d = \sqrt{4} \times \sqrt{2}$$ $$d = 2\sqrt{2}$$

The length of the straight line segment AB is $\boxed{2\sqrt{2}}$.

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: Functions

Given functions:

$$f(x) = 2x + 1$$ $$g(x) = \frac{3}{x-2}$$