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### Section A: Operations on Polynomials

#### 1. \( (2x^2 + 3x - 7) + (3x^2 - 4x - 10) \)

Combine like terms:
\[
(2x^2 + 3x^2) + (3x - 4x) + (-7 - 10) = 5x^2 - x - 17
\]
Final result: \( 5x^2 - x - 17 \)

#### 2. \( (5x^2 - 12x + 1) - (2x^2 + 3x - 7) \)

Distribute the negative sign and combine like terms:
\[
5x^2 - 12x + 1 - 2x^2 - 3x + 7 = (5x^2 - 2x^2) + (-12x - 3x) + (1 + 7) = 3x^2 - 15x + 8
\]
Final result: \( 3x^2 - 15x + 8 \)

#### 3. \( -2x(x^2 + 3x - 8) \)

Distribute \( -2x \):
\[
-2x(x^2) - 2x(3x) - 2x(-8) = -2x^3 - 6x^2 + 16x
\]
Final result: \( -2x^3 - 6x^2 + 16x \)

#### 4. \( (x^2 + 3x - 4)(x^2 - 6x + 5) \)

Expand the product using distribution (FOIL method):
\[
(x^2 + 3x - 4)(x^2 - 6x + 5) = x^2(x^2 - 6x + 5) + 3x(x^2 - 6x + 5) - 4(x^2 - 6x + 5)
\]
Expanding:
\[
= x^4 - 6x^3 + 5x^2 + 3x^3 - 18x^2 + 15x - 4x^2 + 24x - 20
\]
Combine like terms:
\[
x^4 - 3x^3 - 17x^2 + 39x - 20
\]
Final result: \( x^4 - 3x^3 - 17x^2 + 39x - 20 \)

---

### Section B: Composite Functions

Given: \( f(x) = -3x + 7 \) and \( g(x) = 2x^2 - 8 \)

#### 1. \( (g \circ f)(x) \)

This means \( g(f(x)) \). Substituting \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(-3x + 7) = 2(-3x + 7)^2 - 8
\]
First, expand \( (-3x + 7)^2 \):
\[
(-3x + 7)^2 = 9x^2 - 42x + 49
\]
Substitute back:
\[
g(f(x)) = 2(9x^2 - 42x + 49) - 8 = 18x^2 - 84x + 98 - 8 = 18x^2 - 84x + 90
\]
Final result: \( 18x^2 - 84x + 90 \)

#### 2. \( f(g(x)) \)

This means \( f(g(x)) \). Substituting \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(2x^2 - 8) = -3(2x^2 - 8) + 7
\]
Expand:
\[
f(g(x)) = -6x^2 + 24 + 7 = -6x^2 + 31
\]
Final result: \( -6x^2 + 31 \)

---

Would you like further details or explanations on any of these steps?

Here are five related questions:
1. Can you apply the distributive property to other binomial products?
2. How do you simplify polynomial division?
3. What is the general approach for composite functions?
4. How do you identify the degree of a polynomial?
5. What are some real-world applications of composite functions?

**Tip:** Always simplify expressions fully before substituting in composite functions for cleaner results.

Operations on polynomials and composite functions as shown in the uploaded image.

Solve polynomial operations and composite functions using key algebraic techniques. This page covers addition, subtraction, and multiplication of polynomials, as well as composite functions with detailed step-by-step solutions. Perfect for students in Grades 9-10.

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