https://file.aiquickdraw.com/mathbot/file/1725992929314val3a5a3.jpg

I have reviewed the image and will now solve the indicated mathematical problems.

### Part 1: Perform the operations and simplify
(a) \( 3(x + 6) + 4(2x - 5) \)
\[
= 3x + 18 + 8x - 20
= (3x + 8x) + (18 - 20)
= 11x - 2
\]

(b) \( (x + 3)(4x - 5) \)
\[
= x(4x - 5) + 3(4x - 5)
= 4x^2 - 5x + 12x - 15
= 4x^2 + 7x - 15
\]

(c) \( (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) \)
\[
= (\sqrt{a})^2 - (\sqrt{b})^2
= a - b
\]

(d) \( (2x + 3)^2 \)
\[
= (2x + 3)(2x + 3)
= 4x^2 + 6x + 6x + 9
= 4x^2 + 12x + 9
\]

(e) \( (x + 2)^3 \)
\[
= (x + 2)(x + 2)(x + 2)
= (x^2 + 4x + 4)(x + 2)
= x(x^2 + 4x + 4) + 2(x^2 + 4x + 4)
= x^3 + 4x^2 + 4x + 2x^2 + 8x + 8
= x^3 + 6x^2 + 12x + 8
\]

### Part 2: Factor each expression completely
(a) \( 4x^2 - 25 \)
This is a difference of squares:
\[
= (2x + 5)(2x - 5)
\]

(b) \( 2x^2 + 5x - 12 \)
We find factors of \( -24 \) that add to \( 5 \), which are \( 8 \) and \( -3 \):
\[
= 2x^2 + 8x - 3x - 12
= 2x(x + 4) - 3(x + 4)
= (2x - 3)(x + 4)
\]

(c) \( x^3 - 3x^2 - 4x + 12 \)
Group the terms:
\[
= (x^3 - 3x^2) - (4x - 12)
= x^2(x - 3) - 4(x - 3)
= (x^2 - 4)(x - 3)
\]
Factor \( x^2 - 4 \) as a difference of squares:
\[
= (x + 2)(x - 2)(x - 3)
\]

Let me know if you need further clarification, and here are some additional practice questions:

1. Simplify \( 5(2x - 4) + 3(3x + 1) \).
2. Factor \( 9x^2 - 16 \).
3. Expand and simplify \( (3x - 2)(x^2 + 4x + 1) \).
4. Factor \( 3x^2 + 14x + 8 \).
5. Solve \( (x + 5)^2 = 49 \).

**Tip**: For factoring quadratic expressions, try splitting the middle term to simplify the process.

Perform the indicated operations and simplify:
(a) 3(x + 6) + 4(2x − 5)
(b) (x + 3)(4x − 5)
(c) (√a + √b)(√a − √b)
(d) (2x + 3)^2
(e) (x + 2)^3
Factor each expression completely:
(a) 4x^2 − 25
(b) 2x^2 + 5x − 12
(c) x^3 − 3x^2 − 4x + 12

Learn how to simplify and factor algebraic expressions involving polynomials and square roots. This step-by-step guide covers polynomial expansion, factoring, and the difference of squares method. Suitable for Grades 9-10.

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