I see you've uploaded a math worksheet. Let me analyze and solve the problems for you. I'll break them into sections for clarity:

3. Multiply and simplify

Let’s solve each part step by step:

(a) $3x^2(2x - 4)$

Distribute $3x^2$ across $(2x - 4)$: $3x^2 \cdot 2x = 6x^3, \quad 3x^2 \cdot (-4) = -12x^2$ Final result: $6x^3 - 12x^2$

(b) $(x + 2)(2x^2 - 3x + 5)$

Expand $(x + 2) \cdot (2x^2 - 3x + 5)$: $x \cdot 2x^2 = 2x^3, \quad x \cdot (-3x) = -3x^2, \quad x \cdot 5 = 5x$ $2 \cdot 2x^2 = 4x^2, \quad 2 \cdot (-3x) = -6x, \quad 2 \cdot 5 = 10$ Combine like terms: $2x^3 + (-3x^2 + 4x^2) + (5x - 6x) + 10 = 2x^3 + x^2 - x + 10$

(c) $(r + 4)(r - 4)$

Recognize this as a difference of squares: $(r + 4)(r - 4) = r^2 - 16$

(d) $(x + 4)(x - 2)(x + 3)$

Step 1: Expand $(x + 4)(x - 2)$: $x^2 - 2x + 4x - 8 = x^2 + 2x - 8$ Step 2: Multiply $(x^2 + 2x - 8)$ by $(x + 3)$: $x \cdot x^2 = x^3, \quad x \cdot 2x = 2x^2, \quad x \cdot (-8) = -8x$ $3 \cdot x^2 = 3x^2, \quad 3 \cdot 2x = 6x, \quad 3 \cdot (-8) = -24$ Combine like terms: $x^3 + (2x^2 + 3x^2) + (-8x + 6x) - 24 = x^3 + 5x^2 - 2x - 24$

(e) $(x^2 + 5)(x^2 - 2x + 5)$

Expand $(x^2 + 5)(x^2 - 2x + 5)$: $x^2 \cdot x^2 = x^4, \quad x^2 \cdot (-2x) = -2x^3, \quad x^2 \cdot 5 = 5x^2$ $5 \cdot x^2 = 5x^2, \quad 5 \cdot (-2x) = -10x, \quad 5 \cdot 5 = 25$ Combine like terms: $x^4 - 2x^3 + (5x^2 + 5x^2) - 10x + 25 = x^4 - 2x^3 + 10x^2 - 10x + 25$

(f) $(2x - 3)^2$

This is a square of a binomial: $(2x - 3)^2 = (2x)^2 - 2(2x)(3) + 3^2$ $4x^2 - 12x + 9$

Review Questions

I. Equation: $3x + 27 = a(x + 9)$

If there are infinitely many solutions, the left and right sides must be equivalent. Expand $a(x + 9)$: $a(x + 9) = ax + 9a$ Equating coefficients of $x$ and constants: $3 = a, \quad 27 = 9a$ Solve $9a = 27$: $a = 3$

II. Function $f(x) = 6x - 30$, find x-intercept

The x-intercept occurs when $f(x) = 0$: $6x - 30 = 0$ Solve for $x$: $6x = 30 \quad \Rightarrow \quad x = 5$ The x-intercept is $(5, 0)$.

Correct answer: $\boxed{D. \ (5, 0)}$

Do you need further clarification or details?

Here are 5 questions to extend your understanding:

1. What are the properties of the difference of squares, and how are they applied here?
2. Why is the distributive property important in expanding polynomials?
3. How do you identify the coefficients for infinite solutions in equations?
4. What is the geometric meaning of the x-intercept in a linear function?
5. Can you verify the results of part (e) using another method like substitution?

Tip: When simplifying polynomials, always organize terms by degree to avoid missing like terms during addition.