You are given the following two functions:

\[
f(x) = 2x + 3
\]
\[
g(x) = x^2 - x + 2
\]
Additionally, you are provided that \( g(a) = 4 \), and you need to find \( (g \circ f)(x) \), which is the composition of \( g \) and \( f \), denoted by \( gof \). 

To solve \( (g \circ f)(x) \), we first substitute \( f(x) \) into \( g(x) \):

\[
(g \circ f)(x) = g(f(x))
\]

1. First, find \( f(x) \):
\[
f(x) = 2x + 3
\]

2. Now, substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(2x + 3) = (2x + 3)^2 - (2x + 3) + 2
\]

3. Simplify each part:

- Expand \( (2x + 3)^2 \):
\[
(2x + 3)^2 = 4x^2 + 12x + 9
\]

- Now substitute it into \( g(f(x)) \):
\[
g(f(x)) = (4x^2 + 12x + 9) - (2x + 3) + 2
\]
\[
= 4x^2 + 12x + 9 - 2x - 3 + 2
\]
\[
= 4x^2 + 10x + 8
\]

Thus, \( (g \circ f)(x) = 4x^2 + 10x + 8 \).

### Now, regarding \( g(a) = 4 \):

This means:
\[
g(a) = a^2 - a + 2 = 4
\]

Solving for \( a \):

\[
a^2 - a + 2 = 4
\]
\[
a^2 - a - 2 = 0
\]

Factor the quadratic equation:
\[
(a - 2)(a + 1) = 0
\]

So, \( a = 2 \) or \( a = -1 \).

Would you like more details on the composition or solving the equation?

### Related Questions:

1. What is the meaning of function composition in mathematics?
2. How do you expand the square of a binomial?
3. Can a function have more than one value of \( x \) that satisfies \( g(x) = 4 \)?
4. How do you factor a quadratic equation?
5. What is the domain of the composite function \( (g \circ f)(x) \)?

### Tip:
Always check the domain of the inner function when performing function composition to ensure the result is valid for all \( x \).

f(x) = 2x + 3, g(x) = x² - x + 2, g(a) = 4, gof?

Explore how to solve the composition of functions g(f(x)) where f(x) = 2x + 3 and g(x) = x² - x + 2. Learn key algebraic techniques such as binomial expansion and factoring quadratic equations. Suitable for students in Grades 10-12.

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