The problem provides two functions, $f(x) = -x^2 + 4$ and $g(x) = 4x + 4$, and asks us to solve a series of inequalities and equations involving these functions. Let's go through each part step-by-step:

(a) Solve $f(x) = 0$

To solve $f(x) = 0$:

$f(x) = -x^2 + 4 = 0$

$-x^2 + 4 = 0 \implies x^2 = 4$

$x = \pm 2$

(b) Solve $g(x) = 0$

To solve $g(x) = 0$:

$g(x) = 4x + 4 = 0$

$4x + 4 = 0 \implies 4x = -4$

$x = -1$

(c) Solve $f(x) = g(x)$

To solve $f(x) = g(x)$:

$-x^2 + 4 = 4x + 4$

Simplify:

$-x^2 - 4x = 0$

Factor out:

$-x(x + 4) = 0$

$x = 0 \text{ or } x = -4$

(d) Solve $f(x) > 0$

Solve $-x^2 + 4 > 0$:

$x^2 < 4 \implies -2 < x < 2$

(e) Solve $g(x) \leq 0$

Solve $4x + 4 \leq 0$:

$4x \leq -4 \implies x \leq -1$

(f) Solve $f(x) > g(x)$

Solve $-x^2 + 4 > 4x + 4$:

$-x^2 - 4x < 0$

$x(x + 4) > 0$

Interval test gives:

$x < -4 \text{ or } x > 0$

(g) Solve $f(x) \geq 4$

Solve $-x^2 + 4 \geq 4$:

$x^2 \leq 0$

$x = 0$

Would you like further details or have any questions?

Related Questions:

1. How do you solve inequalities with quadratic functions?
2. What are the solutions to a function intersecting another line?
3. Can $x^2$ have negative values? If not, why?
4. How do you determine where one function is greater than another?
5. What steps should you take when factoring polynomials?

Tip: When solving inequalities, sketching graphs can help visualize the solutions.