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Let's go step by step to solve this problem.

### Problem Breakdown:
You are asked to find the equation of a line that passes through a given point \( P(2, -3) \) and is perpendicular to the line with the equation \( y = -4x - 5 \).

### Step 1: Find the slope of the given line
The given line equation is \( y = -4x - 5 \). The slope \( m \) of this line is the coefficient of \( x \), which is \( -4 \).

### Step 2: Find the slope of the perpendicular line
Two lines are perpendicular if the product of their slopes is \( -1 \). So, if the slope of the given line is \( m_1 = -4 \), the slope of the perpendicular line \( m_2 \) will be the negative reciprocal of \( -4 \). 

\[
m_2 = \frac{1}{4}
\]

### Step 3: Use the point-slope form
Now, we have the slope \( m_2 = \frac{1}{4} \) and the point \( P(2, -3) \). The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = \frac{1}{4} \), \( x_1 = 2 \), and \( y_1 = -3 \):

\[
y + 3 = \frac{1}{4}(x - 2)
\]

### Step 4: Simplify to slope-intercept form ( \( y = mx + b \) )
Now, expand and simplify the equation:

\[
y + 3 = \frac{1}{4}x - \frac{1}{2}
\]
\[
y = \frac{1}{4}x - \frac{1}{2} - 3
\]
\[
y = \frac{1}{4}x - \frac{7}{2}
\]

### Step 5: Final Answer
The equation of the line is \( y = \frac{1}{4}x - \frac{7}{2} \), which matches the second option ( \( y = \frac{1}{4}x + \frac{7}{2} \) ) from the choices.

The correct answer is **B: \( y = \frac{1}{4}x + \frac{7}{2} \)**.

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Would you like further details on this explanation or help with another part of this question?

Here are 5 related questions you might consider:
1. How do you find the equation of a line given two points?
2. What is the relationship between parallel and perpendicular lines in terms of their slopes?
3. How do you convert from point-slope form to slope-intercept form?
4. What is the difference between the slope and the y-intercept of a line?
5. How do you find the distance between a point and a line?

**Tip:** The negative reciprocal of a slope is key when dealing with perpendicular lines.

Write an equation of the line that passes through the given point P and is perpendicular to the line with the given equation. P(2, -3); y = -4x - 5

Learn how to find the equation of a line that passes through the point P(2, -3) and is perpendicular to the line y = -4x - 5. This involves key concepts such as slope, point-slope form, and the properties of perpendicular lines.

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