We are given two functions:

1. $f(x) = x^2 + 6x - 2$
2. $g(x) = 2x - 10$

To find $f(g(x))$, we need to substitute $g(x)$ into $f(x)$.

Step-by-step:

1. First, express $f(g(x))$: $f(g(x)) = f(2x - 10)$ This means that wherever there is an $x$ in $f(x)$, we replace it with $2x - 10$.

2. Now substitute $2x - 10$ into $f(x) = x^2 + 6x - 2$: $f(2x - 10) = (2x - 10)^2 + 6(2x - 10) - 2$

3. Expand the terms: $(2x - 10)^2 = (2x)^2 - 2(2x)(10) + (10)^2 = 4x^2 - 40x + 100$ $6(2x - 10) = 12x - 60$

4. Combine everything: $f(2x - 10) = 4x^2 - 40x + 100 + 12x - 60 - 2$

5. Simplify the expression: $f(2x - 10) = 4x^2 - 28x + 38$

Thus, $f(g(x)) = 4x^2 - 28x + 38$.

Would you like more details or have any questions about this? Here are 5 questions related to this process:

1. How do we simplify quadratic expressions like $(2x - 10)^2$?
2. What is the importance of substituting one function into another in calculus?
3. Can we find the inverse of $f(g(x))$?
4. What happens if we find $g(f(x))$ instead of $f(g(x))$?
5. How can we graph $f(g(x))$ and interpret its behavior?

Tip: When substituting functions, ensure that you correctly handle each variable and expand carefully to avoid errors.