QUESTION 3 Factorise each of the following expressions fully 3.1) $$ 6 a^{2} b^{3} c^{4}-24 a^{3} b^{2} c^{3} $$ 8) $$ 3 x^{2}+6 x-45 $$ 3.3) $$ 2 x(y-3 z)-6 x^{2}(3 z-y) $$ 3.4) $$ (a-2)^{3}-64 $$ (4) 3.2) Fully factorise each of the following expressions: 3.5.1) $$ 8 a^{3}-50 a b^{2} $$ (3) $$ 35.2) x^{3}-\frac{1}{64} $$ 3.5.3) $$ a^{2}+a-2 a b-2 b $$ 3.5.4) $$ 5 x^{2}-27 x-18 $$ (2) (3) (2)

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Factor the expression by following steps:
- step0: Factor:
  $$5x^{2}-27x-18$$
- step1: Rewrite the expression:
  $$5x^{2}+\left(3-30\right)x-18$$
- step2: Calculate:
  $$5x^{2}+3x-30x-18$$
- step3: Rewrite the expression:
  $$x\times 5x+x\times 3-6\times 5x-6\times 3$$
- step4: Factor the expression:
  $$x\left(5x+3\right)-6\left(5x+3\right)$$
- step5: Factor the expression:
  $$\left(x-6\right)\left(5x+3\right)$$
Factor the expression $$ x^{3}-\frac{1}{64} $$.
Factor the expression by following steps:
- step0: Factor:
  $$x^{3}-\frac{1}{64}$$
- step1: Factor the expression:
  $$\frac{1}{64}\left(64x^{3}-1\right)$$
- step2: Factor the expression:
  $$\frac{1}{64}\left(4x-1\right)\left(16x^{2}+4x+1\right)$$
Factor the expression $$ a^{2}+a-2 a b-2 b $$.
Factor the expression by following steps:
- step0: Factor:
  $$a^{2}+a-2ab-2b$$
- step1: Calculate:
  $$a^{2}+a-2ba-2b$$
- step2: Rewrite the expression:
  $$a\times a+a-2ba-2b$$
- step3: Factor the expression:
  $$a\left(a+1\right)-2b\left(a+1\right)$$
- step4: Factor the expression:
  $$\left(a-2b\right)\left(a+1\right)$$
Factor the expression $$ (a-2)^{3}-64 $$.
Factor the expression by following steps:
- step0: Factor:
  $$\left(a-2\right)^{3}-64$$
- step1: Rewrite the expression:
  $$\left(a-2\right)^{3}-4^{3}$$
- step2: Factor the expression:
  $$\left(a-2-4\right)\left(\left(a-2\right)^{2}+\left(a-2\right)\times 4+4^{2}\right)$$
- step3: Evaluate:
  $$\left(a-2-4\right)\left(\left(a-2\right)^{2}+4\left(a-2\right)+16\right)$$
- step4: Calculate:
  $$\left(a-6\right)\left(\left(a-2\right)^{2}+4\left(a-2\right)+16\right)$$
- step5: Calculate:
  $$\left(a-6\right)\left(a^{2}+12\right)$$
Factor the expression $$ 8 a^{3}-50 a b^{2} $$.
Factor the expression by following steps:
- step0: Factor:
  $$8a^{3}-50ab^{2}$$
- step1: Factor the expression:
  $$2a\left(4a^{2}-25b^{2}\right)$$
- step2: Factor the expression:
  $$2a\left(2a-5b\right)\left(2a+5b\right)$$
Factor the expression $$ 3 x^{2}+6 x-45 $$.
Factor the expression by following steps:
- step0: Factor:
  $$3x^{2}+6x-45$$
- step1: Rewrite the expression:
  $$3x^{2}+3\times 2x-3\times 15$$
- step2: Factor the expression:
  $$3\left(x^{2}+2x-15\right)$$
- step3: Factor the expression:
  $$3\left(x-3\right)\left(x+5\right)$$
Factor the expression $$ 6 a^{2} b^{3} c^{4}-24 a^{3} b^{2} c^{3} $$.
Factor the expression by following steps:
- step0: Factor:
  $$6a^{2}b^{3}c^{4}-24a^{3}b^{2}c^{3}$$
- step1: Rewrite the expression:
  $$6a^{2}b^{2}c^{3}bc-6a^{2}b^{2}c^{3}\times 4a$$
- step2: Factor the expression:
  $$6a^{2}b^{2}c^{3}\left(bc-4a\right)$$
Factor the expression $$ 2 x(y-3 z)-6 x^{2}(3 z-y) $$.
Factor the expression by following steps:
- step0: Factor:
  $$2x\left(y-3z\right)-6x^{2}\left(3z-y\right)$$
- step1: Rewrite the expression:
  $$2x\left(-3z+y\right)+2x\left(-3z+y\right)\times 3x$$
- step2: Factor the expression:
  $$2x\left(-3z+y\right)\left(1+3x\right)$$
Here are the fully factored forms of each expression:

### 3.1) $$ 6 a^{2} b^{3} c^{4}-24 a^{3} b^{2} c^{3} $$
**Factored Form:** 
$$
6a^{2}b^{2}c^{3}(bc-4a)
$$

### 3.2) $$ 3 x^{2}+6 x-45 $$
**Factored Form:** 
$$
3(x-3)(x+5)
$$

### 3.3) $$ 2 x(y-3 z)-6 x^{2}(3 z-y) $$
**Factored Form:** 
$$
2x(-3z+y)(1+3x)
$$

### 3.4) $$ (a-2)^{3}-64 $$
**Factored Form:** 
$$
(a-6)(a^{2}+12)
$$

### 3.5.1) $$ 8 a^{3}-50 a b^{2} $$
**Factored Form:** 
$$
2a(2a-5b)(2a+5b)
$$

### 3.5.2) $$ x^{3}-\frac{1}{64} $$
**Factored Form:** 
$$
\frac{1}{64}(4x-1)(16x^{2}+4x+1)
$$

### 3.5.3) $$ a^{2}+a-2 a b-2 b $$
**Factored Form:** 
$$
(a-2b)(a+1)
$$

### 3.5.4) $$ 5 x^{2}-27 x-18 $$
**Factored Form:** 
$$
(x-6)(5x+3)
$$

These factorizations are fully simplified and represent the expressions in their factored forms.
Here are the fully factored forms of each expression: 1. $$ 6 a^{2} b^{3} c^{4}-24 a^{3} b^{2} c^{3} = 6a^{2}b^{2}c^{3}(bc-4a) $$ 2. $$ 3 x^{2}+6 x-45 = 3(x-3)(x+5) $$ 3. $$ 2 x(y-3 z)-6 x^{2}(3 z-y) = 2x(-3z+y)(1+3x) $$ 4. $$ (a-2)^{3}-64 = (a-6)(a^{2}+12) $$ 5. $$ 8 a^{3}-50 a b^{2} = 2a(2a-5b)(2a+5b) $$ 6. $$ x^{3}-\frac{1}{64} = \frac{1}{64}(4x-1)(16x^{2}+4x+1) $$ 7. $$ a^{2}+a-2 a b-2 b = (a-2b)(a+1) $$ 8. $$ 5 x^{2}-27 x-18 = (x-6)(5x+3) $$

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