Answer
$x=-7$ or $x=-3$
Work Step by Step
The exercise is asking us to identify the zeros of $f(g(x))$:
$$f(g(x)) = 4 - (x + 5)^{2}$$
$$f(g(x)) = 4 - (x^{2} + 10x + 25)$$
$$f(g(x)) = 4 - x^{2} - 10x - 25$$
$$f(g(x)) = -x^{2} - 10x - 21$$
Finding the zeros of this function:
$$f(g(x)) = 0 = -x^{2} - 10x - 21$$which is the same as:
$$0 = x^{2} + 10x + 21$$
Since two factors of 21 that, when added, give 10 are 3 and 7, we can factorize the function into:
$$0 = (x+ 7)(x+3)$$ and solve:
$0 = x+7$ or $0=x+3$
$0-7=x$ or $0-3=x$
$-7=x$ or $-3=x$
