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We are asked to find \( (g \circ f)(x) \), which represents the composition of the functions \( g(x) \) and \( f(x) \). 

Given:
- \( f(x) = x - 7 \)
- \( g(x) = 2x^2 + 3x - 5 \)

The composition \( (g \circ f)(x) \) means we substitute \( f(x) \) into \( g(x) \). In other words, we replace every occurrence of \( x \) in \( g(x) \) with \( f(x) \).

### Steps:
1. Start by writing \( g(f(x)) \). Since \( g(x) = 2x^2 + 3x - 5 \), substitute \( f(x) = x - 7 \) into \( g(x) \).

\[
g(f(x)) = 2(f(x))^2 + 3(f(x)) - 5
\]

2. Substitute \( f(x) = x - 7 \) into this expression:

\[
g(f(x)) = 2(x - 7)^2 + 3(x - 7) - 5
\]

3. Now, expand each term:
   - Expand \( (x - 7)^2 = x^2 - 14x + 49 \)
   - Expand \( 3(x - 7) = 3x - 21 \)

4. Substitute the expanded terms back:

\[
g(f(x)) = 2(x^2 - 14x + 49) + 3x - 21 - 5
\]

5. Distribute the 2 in the first term:

\[
g(f(x)) = 2x^2 - 28x + 98 + 3x - 21 - 5
\]

6. Combine like terms:

\[
g(f(x)) = 2x^2 - 25x + 72
\]

So, the polynomial in standard form is:

\[
(g \circ f)(x) = 2x^2 - 25x + 72
\]

Would you like more details or have any questions?

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Here are five questions to extend your understanding:

1. What is the general formula for composing two functions \( g(f(x)) \)?
2. How would the process change if \( f(x) \) were a more complicated polynomial?
3. What if you were asked to find \( (f \circ g)(x) \)? How would you proceed?
4. How does function composition relate to transformations in geometry?
5. Can you compose three functions, and how would you approach that?

**Tip:** When expanding quadratic expressions like \( (x - 7)^2 \), remember to double-check each term, especially the middle term from the binomial expansion.

Express your answer as a polynomial in standard form. Given: f(x) = x - 7 and g(x) = 2x^2 + 3x - 5. Find: (g ∘ f)(x).

This problem involves composing two functions, f(x) = x - 7 and g(x) = 2x^2 + 3x - 5, and expressing the result as a polynomial. The step-by-step solution covers expanding binomials and combining like terms. Suitable for students in Grades 9-11.

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