To find $(f \circ g)(x)$, which represents $f(g(x))$, we need to substitute $g(x)$ into $f(x)$.

Given: $f(x) = 2x^2 + 5x + 15$ $g(x) = -x + 5$

Step 1: Substitute $g(x)$ into $f(x)$

Since $f(g(x)) = f(-x + 5)$, replace every $x$ in $f(x)$ with $-x + 5$.

Thus, $f(-x + 5) = 2(-x + 5)^2 + 5(-x + 5) + 15$

Step 2: Expand $(-x + 5)^2$

$(-x + 5)^2 = x^2 - 10x + 25$ So, $f(-x + 5) = 2(x^2 - 10x + 25) + 5(-x + 5) + 15$

Step 3: Distribute and Simplify

Expanding each term: $= 2x^2 - 20x + 50 - 5x + 25 + 15$

Combine like terms: $= 2x^2 - 25x + 90$

Final Answer

$(f \circ g)(x) = 2x^2 - 25x + 90$

Would you like further details on any specific step?

Related Questions:

1. What is the process of function composition and how does it work?
2. How would the result change if $g(x) = x + 5$ instead?
3. What is the significance of writing a polynomial in standard form?
4. How can you verify that $(f \circ g)(x)$ is correct?
5. What are other ways to represent compositions of functions?

Tip: When composing functions, remember to replace the variable consistently to avoid errors during substitution and simplification.