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.