Let me analyze the math problems in your image. I’ll extract the details and solve each one step by step.### Transcribed Problems

The image contains the following math problems:

Section 1: Perform the Calculations

1. $(x^2 - 7x + 8) - 3(x^2 + 2x - 3)$
2. $\frac{6x - 5 - x}{7 - 2}$

Section 2: Simplify or Solve

1. $9a^2b \div (-30) \times 46b$
2. $4\sqrt{3} - \sqrt{12} + \sqrt{27}$

Section 3: Expand

1. $(x + 1)(3x^2 - 3x + 5)$

Now, I'll solve each question step by step.

Solutions

1.1 $(x^2 - 7x + 8) - 3(x^2 + 2x - 3)$

Expand and simplify: $(x^2 - 7x + 8) - 3(x^2 + 2x - 3)$ $= x^2 - 7x + 8 - 3x^2 - 6x + 9$ Combine like terms: $x^2 - 3x^2 = -2x^2,\ \ -7x - 6x = -13x,\ \ 8 + 9 = 17$ Final result: $-2x^2 - 13x + 17$

1.2 $\frac{6x - 5 - x}{7 - 2}$

Simplify numerator and denominator: $\text{Numerator: } 6x - 5 - x = 5x - 5$ $\text{Denominator: } 7 - 2 = 5$ Simplify fraction: $\frac{5x - 5}{5} = x - 1$

2.1 $9a^2b \div (-30) \times 46b$

Simplify step by step: $9a^2b \div (-30) = \frac{9a^2b}{-30} = -\frac{3a^2b}{10}$ Multiply by $46b$: $-\frac{3a^2b}{10} \times 46b = -\frac{138a^2b^2}{10} = -13.8a^2b^2$

2.2 $4\sqrt{3} - \sqrt{12} + \sqrt{27}$

Simplify each square root: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3},\ \ \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}$ Substitute: $4\sqrt{3} - \sqrt{12} + \sqrt{27} = 4\sqrt{3} - 2\sqrt{3} + 3\sqrt{3}$ Combine terms: $(4 - 2 + 3)\sqrt{3} = 5\sqrt{3}$

3.1 $(x + 1)(3x^2 - 3x + 5)$

Expand using distributive property: $(x + 1)(3x^2 - 3x + 5) = x(3x^2 - 3x + 5) + 1(3x^2 - 3x + 5)$ $= 3x^3 - 3x^2 + 5x + 3x^2 - 3x + 5$ Combine like terms: $3x^3 - 3x^2 + 3x^2 + 5x - 3x + 5 = 3x^3 + 2x + 5$

Final Answers

1.1. $-2x^2 - 13x + 17$
1.2. $x - 1$
2.1. $-13.8a^2b^2$
2.2. $5\sqrt{3}$
3.1. $3x^3 + 2x + 5$

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Related Questions

1. How do you simplify algebraic expressions effectively?
2. What are the common mistakes in fraction simplification?
3. How do you handle multiplication of terms with variables?
4. Can square roots always be simplified?
5. How do you verify if polynomial expansions are correct?

Tip: Always double-check your expanded terms to ensure proper combination of like terms!