https://file.aiquickdraw.com/mathbot/file/1726066544592cvf1rl47.jpg

Let's solve the given problem step by step.

The functions provided are:
\[
f(x) = -2x^2 - 2x - 4
\]
\[
g(x) = -x - 5
\]

### (a) \( f(5) + g(5) = -74 \)
Let's verify this first:

1. **Calculate \( f(5) \):**
   \[
   f(5) = -2(5)^2 - 2(5) - 4 = -2(25) - 10 - 4 = -50 - 10 - 4 = -64
   \]

2. **Calculate \( g(5) \):**
   \[
   g(5) = -(5) - 5 = -5 - 5 = -10
   \]

3. **Sum \( f(5) + g(5) \):**
   \[
   f(5) + g(5) = -64 + (-10) = -74
   \]
   The result matches the given answer of \(-74\).

### (b) \( f(7) \cdot g(7) = 1392 \)
Let's verify this:

1. **Calculate \( f(7) \):**
   \[
   f(7) = -2(7)^2 - 2(7) - 4 = -2(49) - 14 - 4 = -98 - 14 - 4 = -116
   \]

2. **Calculate \( g(7) \):**
   \[
   g(7) = -(7) - 5 = -7 - 5 = -12
   \]

3. **Product \( f(7) \cdot g(7) \):**
   \[
   f(7) \cdot g(7) = -116 \times -12 = 1392
   \]
   The result matches the given answer of 1392.

### (c) \( f(g(4)) = ? \)
Let's find this:

1. **Calculate \( g(4) \):**
   \[
   g(4) = -(4) - 5 = -4 - 5 = -9
   \]

2. **Calculate \( f(g(4)) = f(-9) \):**
   \[
   f(-9) = -2(-9)^2 - 2(-9) - 4 = -2(81) + 18 - 4 = -162 + 18 - 4 = -148
   \]
   So, \( f(g(4)) = -148 \).

### (d) \( g(f(5)) = 59 \)
Let's find this:

1. **We already calculated \( f(5) \) earlier:**
   \[
   f(5) = -64
   \]

2. **Calculate \( g(f(5)) = g(-64) \):**
   \[
   g(-64) = -(-64) - 5 = 64 - 5 = 59
   \]
   This matches the given answer of 59.

### Summary:
- (a) \( f(5) + g(5) = -74 \)
- (b) \( f(7) \cdot g(7) = 1392 \)
- (c) \( f(g(4)) = -148 \)
- (d) \( g(f(5)) = 59 \)

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

1. What would be \( f(g(2)) \)?
2. How do the graphs of \( f(x) \) and \( g(x) \) compare?
3. What is the derivative of \( f(x) \)?
4. How would you solve \( f(x) = 0 \)?
5. What would happen if you change \( g(x) \) to a different linear function?

**Tip:** When substituting values into functions, always perform the operations step by step to avoid errors.

Let f(x) = -2x^2 - 2x - 4. Let g(x) = -x - 5. Find the following: (a) f(5) + g(5), (b) f(7) * g(7), (c) f(g(4)), (d) g(f(5))

Solve function problems with f(x) = -2x^2 - 2x - 4 and g(x) = -x - 5, including calculating f(5) + g(5), f(7) * g(7), f(g(4)), and g(f(5)). These problems cover quadratic and linear functions, suitable for Grade 9-10 students.

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