To compute $(f \circ f)(x)$, it means applying the function $f(x)$ to itself. In other words, we substitute $f(x)$ into $f(x)$.

Step 1: Recall the given function $f(x)$

$f(x) = x^2 + 4x + 10$

Step 2: Set up $(f \circ f)(x)$

This means we want to find $f(f(x))$. We substitute $f(x)$ into $f$.

$f(f(x)) = f(x^2 + 4x + 10)$

Step 3: Apply the function $f$ to $x^2 + 4x + 10$

Since $f(x) = x^2 + 4x + 10$, we substitute $x^2 + 4x + 10$ wherever there is an $x$ in the formula for $f(x)$:

$f(f(x)) = (x^2 + 4x + 10)^2 + 4(x^2 + 4x + 10) + 10$

Step 4: Simplify the expression

First, we need to expand $(x^2 + 4x + 10)^2$:

$(x^2 + 4x + 10)^2 = x^4 + 8x^3 + 28x^2 + 80x + 100$

Next, expand $4(x^2 + 4x + 10)$:

$4(x^2 + 4x + 10) = 4x^2 + 16x + 40$

Now, combine everything:

$f(f(x)) = x^4 + 8x^3 + 28x^2 + 80x + 100 + 4x^2 + 16x + 40 + 10$

Step 5: Combine like terms

$f(f(x)) = x^4 + 8x^3 + (28x^2 + 4x^2) + (80x + 16x) + (100 + 40 + 10)$ $f(f(x)) = x^4 + 8x^3 + 32x^2 + 96x + 150$

Thus, the result is: $(f \circ f)(x) = x^4 + 8x^3 + 32x^2 + 96x + 150$

Would you like more details on any of the steps? Here are some related questions:

1. What happens when you compose $g(x)$ with itself?
2. How does function composition differ from function addition?
3. Can you explain how to factor quadratic functions?
4. How would you graph $f(f(x))$?
5. What does the behavior of higher-degree polynomials look like?

Tip: When composing functions, ensure each function is substituted into every instance of the variable carefully to avoid missing terms.