https://file.aiquickdraw.com/mathbot/file/1726610889891tti27gi0.jpg

Let's go through the math problems shown in the image:

**8. What is the complete factored form of \(4a^2 - 4a^2b + a^2b^2\)?**

We have \(4a^2 - 4a^2b + a^2b^2\). This looks like a perfect square trinomial:
\[
(2a - ab)^2
\]
So, the correct answer is **C**.

**9. What is the complete factored form of \(8x^2 - 24xy + 18y^2\)?**

This expression is a trinomial and can be factored as:
\[
2(2x - 3y)(2x - 3y) = 2(2x - 3y)^2
\]
So, the correct answer is **B**.

**10. Which of the following trinomials is factorable?**

For the options:
- \(x^2 - 6x + 7\) (not factorable over integers)
- \(2x^2 + 5x + 10\) (not factorable)
- \(x^2 + 3x + 2\) (factorable as \((x + 1)(x + 2)\))
- \(3x^2 - 6x + 12\) (not factorable)

The correct answer is **C**.

**11. All of the following trinomials are factorable, EXCEPT one. What is it?**

We need to factor each option to find the exception:
- \(x^2 - 2x - 3\) factors as \((x - 3)(x + 1)\)
- \(x^2 + 5x + 6\) factors as \((x + 2)(x + 3)\)
- \(2x^2 + 3x - 2\) factors as \((2x - 1)(x + 2)\)
- \(2x^2 + 2x + 4\) is not factorable.

The correct answer is **D**.

**12. What is the complete factored form of \(x^2 - 4x - 96\)?**

To factor \(x^2 - 4x - 96\), find factors of \(-96\) that add up to \(-4\):
\[
x^2 - 4x - 96 = (x - 12)(x + 8)
\]
So, the correct answer is **B**.

**13. What is the complete factored form of \(2x^2 - 6x - 8\)?**

Factor out 2 first:
\[
2(x^2 - 3x - 4) = 2(x - 4)(x + 1)
\]
The correct answer is **A**.

**14. Which of the following trinomials has factors \((3a - 2b)(2a - b)\)?**

Multiply the factors:
\[
(3a - 2b)(2a - b) = 6a^2 - 3ab - 4ab + 2b^2 = 6a^2 - 7ab + 2b^2
\]
The correct answer is **D**.

**15. A rectangular garden has an area \(6x^2 + x - 2\) square meters. If the length is represented by \(3x + 2\), find a binomial that represents the width.**

To find the width, divide the area by the length:
\[
\frac{6x^2 + x - 2}{3x + 2} = 2x - 1
\]
The correct answer is **C**.

Would you like to go over any of the solutions in more detail?

Here are five related questions for practice:
1. Factor \(3x^2 - 12x + 9\).
2. Solve \(x^2 - 9 = 0\).
3. Factor \(4x^2 + 12x + 9\).
4. If the area of a rectangle is \(x^2 + 7x + 12\) and the length is \(x + 3\), find the width.
5. Factor \(x^2 - 4x + 4\).

**Tip:** Always check if you can factor out the greatest common factor (GCF) before factoring trinomials.

Solve various algebraic problems, including factoring trinomials, perfect squares, and determining the width of a rectangular garden given the area and length.

Practice factoring trinomials, solving quadratic equations, and determining dimensions of rectangles from given areas. Learn key algebraic techniques with step-by-step solutions for problems involving expressions like x^2 - 4x - 96 and 2x^2 - 6x - 8.

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