1. Factorise 1tx $$ x^{2}-3 x+2 $$ $$ 1.2 x^{2}-4 x $$ $$ 1.3 x^{2} 4 $$

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Sure, let's factorize each of the given quadratic expressions step by step.

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### **1.1. Factorize $$ x^{2} - 3x + 2 $$**

**Step 1:** Identify two numbers that multiply to **+2** (the constant term) and add up to **-3** (the coefficient of the middle term).

- The numbers **-1** and **-2** satisfy this condition because:
  - $$-1 \times -2 = +2$$
  - $$-1 + (-2) = -3$$

**Step 2:** Rewrite the quadratic expression using these numbers:
$$
x^{2} - 3x + 2 = (x - 1)(x - 2)
$$

**Answer:**
$$
x^{2} - 3x + 2 = (x - 1)(x - 2)
$$

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### **1.2. Factorize $$ x^{2} - 4x $$**

**Step 1:** Look for the greatest common factor (GCF) in both terms. Here, the GCF is **x**.

**Step 2:** Factor out **x**:
$$
x^{2} - 4x = x(x - 4)
$$

**Answer:**
$$
x^{2} - 4x = x(x - 4)
$$

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### **1.3. Factorize $$ x^{2} + 4 $$**

The expression $$ x^{2} + 4 $$ is a **sum of squares**, which doesn't factorize over the set of real numbers. However, it can be factorized over the set of complex numbers.

**Over Real Numbers:**

- **No Real Factors:** 
  $$
  x^{2} + 4 \quad \text{cannot be factored further}
  $$

**Over Complex Numbers:**

- **Using Imaginary Units:**
  $$
  x^{2} + 4 = (x + 2i)(x - 2i)
  $$
  where $$ i $$ is the imaginary unit, defined by $$ i^{2} = -1 $$.

**Answer:**

- **Over Real Numbers:**
  $$
  x^{2} + 4 \quad \text{cannot be factorized further}
  $$

- **Over Complex Numbers:**
  $$
  x^{2} + 4 = (x + 2i)(x - 2i)
  $$

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If the third expression was meant to be $$ x^{2} - 4 $$ instead of $$ x^{2} + 4 $$, then it can be factorized as follows:

### **Alternative for $$ x^{2} - 4 $$:**

**Step 1:** Recognize this as a **difference of squares**.

**Step 2:** Apply the formula $$ a^{2} - b^{2} = (a - b)(a + b) $$:
$$
x^{2} - 4 = (x - 2)(x + 2)
$$

**Answer:**
$$
x^{2} - 4 = (x - 2)(x + 2)
$$

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Feel free to let me know if you need further clarification or assistance with other problems!
1. $$ x^{2} - 3x + 2 = (x - 1)(x - 2) $$ 2. $$ x^{2} - 4x = x(x - 4) $$ 3. $$ x^{2} + 4 $$ cannot be factored over real numbers but can be factored as $$ (x + 2i)(x - 2i) $$ over complex numbers.

1. Factorise 1tx \( x^{2}-3 x+2 \) \( 1.2 x^{2}-4 x \) \( 1.3 x^{2} 4 \)

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