To solve the equation \( 10 - x^2 = x + 4 \), we can rearrange it as \( 10 - x^2 - x - 4 = 0 \), leading to:
\[ -x^2 - x + 6 = 0 \]
Or by multiplying through by -1:
\[ x^2 + x - 6 = 0 \]
Next, we can factor this quadratic equation:
\[ (x + 3)(x - 2) = 0 \]
Setting each factor to zero gives us the solutions:
- \( x + 3 = 0 \) ➔ \( x = -3 \)
- \( x - 2 = 0 \) ➔ \( x = 2 \)
Thus, the intersection points of \( f(x) = 10 - x^2 \) and \( g(x) = x + 4 \) are \( x = -3 \) and \( x = 2 \).
Regarding the options you provided, here's the summary:
- g(x) = x + 4, f(x) = 10 - x^2, {-3, 2}: This matches our problem, as the functions and solution set is correct.
The other options are incorrect because:
- g(x) = x + 4, f(x) = 10 + x^2, {-3, 1}: Incorrect functions and roots.
- g(x) = x + 4, f(x) = 10 + x^2, {-3, 2}: Incorrect functions; \( f(x) \) should be \( 10 - x^2 \).
Therefore, the correct pair of functions with their intersection points is:
\[ g(x) = x + 4, f(x) = 10 - x^2, \text{ intersection points } {-3, 2} \]