1. Determine the product of the following and simplify fully: $$ (x-2)\left(x^{2}+5 x-1\right) $$ 2. Factorise $$ y^{2}(y-2)+x^{2}(2-y) $$ fully QUESTION 3 [10 Marks] Simplify the following expressions fully: 1. $$ \frac{x^{2}-4}{2 x^{2}+5 x+2} \div \frac{x^{3}-8}{6 x+3} $$ 2. $$ \frac{x}{x+y}+\frac{x^{2}}{y^{2}-x^{2}} $$

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Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{x}{\left(x+y\right)}\right)+\left(\frac{x^{2}}{\left(y^{2}-x^{2}\right)}\right)$$
- step1: Remove the parentheses:
  $$\left(\frac{x}{x+y}\right)+\left(\frac{x^{2}}{y^{2}-x^{2}}\right)$$
- step2: Remove the parentheses:
  $$\frac{x}{x+y}+\left(\frac{x^{2}}{y^{2}-x^{2}}\right)$$
- step3: Remove the parentheses:
  $$\frac{x}{x+y}+\frac{x^{2}}{y^{2}-x^{2}}$$
- step4: Factor the expression:
  $$\frac{x}{x+y}+\frac{x^{2}}{\left(y-x\right)\left(y+x\right)}$$
- step5: Reduce fractions to a common denominator:
  $$\frac{x\left(y-x\right)}{\left(x+y\right)\left(y-x\right)}+\frac{x^{2}}{\left(y-x\right)\left(y+x\right)}$$
- step6: Rewrite the expression:
  $$\frac{x\left(y-x\right)}{\left(y-x\right)\left(y+x\right)}+\frac{x^{2}}{\left(y-x\right)\left(y+x\right)}$$
- step7: Transform the expression:
  $$\frac{x\left(y-x\right)+x^{2}}{\left(y-x\right)\left(y+x\right)}$$
- step8: Multiply the terms:
  $$\frac{xy-x^{2}+x^{2}}{\left(y-x\right)\left(y+x\right)}$$
- step9: Calculate:
  $$\frac{xy}{\left(y-x\right)\left(y+x\right)}$$
- step10: Simplify the product:
  $$\frac{xy}{y^{2}-x^{2}}$$
Factor the expression $$ y^{2}(y-2)+x^{2}(2-y) $$.
Factor the expression by following steps:
- step0: Factor:
  $$y^{2}\left(y-2\right)+x^{2}\left(2-y\right)$$
- step1: Rewrite the expression:
  $$y^{2}\left(y-2\right)-x^{2}\left(y-2\right)$$
- step2: Factor the expression:
  $$\left(y^{2}-x^{2}\right)\left(y-2\right)$$
- step3: Factor the expression:
  $$\left(y+x\right)\left(y-x\right)\left(y-2\right)$$
Calculate or simplify the expression $$ (x^{2}-4)/(2 x^{2}+5 x+2) / ((x^{3}-8)/(6 x+3)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\frac{\left(x^{2}-4\right)}{\left(2x^{2}+5x+2\right)}}{\left(\frac{\left(x^{3}-8\right)}{\left(6x+3\right)}\right)}$$
- step1: Remove the parentheses:
  $$\frac{\frac{x^{2}-4}{2x^{2}+5x+2}}{\frac{x^{3}-8}{6x+3}}$$
- step2: Divide the terms:
  $$\frac{\frac{x-2}{2x+1}}{\frac{x^{3}-8}{6x+3}}$$
- step3: Multiply by the reciprocal:
  $$\frac{x-2}{2x+1}\times \frac{6x+3}{x^{3}-8}$$
- step4: Rewrite the expression:
  $$\frac{x-2}{2x+1}\times \frac{6x+3}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$
- step5: Reduce the fraction:
  $$\frac{1}{2x+1}\times \frac{6x+3}{x^{2}+2x+4}$$
- step6: Rewrite the expression:
  $$\frac{1}{2x+1}\times \frac{3\left(2x+1\right)}{x^{2}+2x+4}$$
- step7: Reduce the fraction:
  $$1\times \frac{3}{x^{2}+2x+4}$$
- step8: Multiply the terms:
  $$\frac{3}{x^{2}+2x+4}$$
Expand the expression $$ (x-2)(x^{2}+5 x-1) $$
Simplify the expression by following steps:
- step0: Multiply the terms:
  $$\left(x-2\right)\left(x^{2}+5x-1\right)$$
- step1: Apply the distributive property:
  $$x\times x^{2}+x\times 5x-x\times 1-2x^{2}-2\times 5x-\left(-2\times 1\right)$$
- step2: Multiply the terms:
  $$x^{3}+5x^{2}-x-2x^{2}-10x-\left(-2\right)$$
- step3: Remove the parentheses:
  $$x^{3}+5x^{2}-x-2x^{2}-10x+2$$
- step4: Subtract the terms:
  $$x^{3}+3x^{2}-11x+2$$
Here are the detailed solutions to your problems:

### Problem 1: Determine the product and simplify
We need to calculate the product of $$ (x-2)(x^{2}+5x-1) $$.

1. **Expand the expression**:
   $$
   (x-2)(x^{2}+5x-1) = x(x^{2}+5x-1) - 2(x^{2}+5x-1)
   $$
   $$
   = x^{3} + 5x^{2} - x - 2x^{2} - 10x + 2
   $$
   $$
   = x^{3} + (5x^{2} - 2x^{2}) + (-x - 10x) + 2
   $$
   $$
   = x^{3} + 3x^{2} - 11x + 2
   $$

**Final Result**: 
$$
x^{3} + 3x^{2} - 11x + 2
$$

### Problem 2: Factorise fully
We need to factor $$ y^{2}(y-2) + x^{2}(2-y) $$.

1. **Rearranging the expression**:
   $$
   y^{2}(y-2) + x^{2}(2-y) = y^{2}(y-2) - x^{2}(y-2)
   $$
   $$
   = (y-2)(y^{2} - x^{2})
   $$
2. **Using the difference of squares**:
   $$
   = (y-2)(y+x)(y-x)
   $$

**Final Result**: 
$$
(y+x)(y-x)(y-2)
$$

### Problem 3: Simplify the following expressions

#### Part 1: 
Simplify $$ \frac{x^{2}-4}{2x^{2}+5x+2} \div \frac{x^{3}-8}{6x+3} $$.

1. **Rewrite the division as multiplication**:
   $$
   = \frac{x^{2}-4}{2x^{2}+5x+2} \cdot \frac{6x+3}{x^{3}-8}
   $$
2. **Factor the expressions**:
   - $$ x^{2}-4 = (x-2)(x+2) $$
   - $$ 2x^{2}+5x+2 = (2x+1)(x+2) $$
   - $$ x^{3}-8 = (x-2)(x^{2}+2x+4) $$
   - $$ 6x+3 = 3(2x+1) $$

3. **Substituting the factors**:
   $$
   = \frac{(x-2)(x+2)}{(2x+1)(x+2)} \cdot \frac{3(2x+1)}{(x-2)(x^{2}+2x+4)}
   $$
4. **Cancel common factors**:
   $$
   = \frac{3}{x^{2}+2x+4}
   $$

**Final Result**: 
$$
\frac{3}{x^{2}+2x+4}
$$

#### Part 2: 
Simplify $$ \frac{x}{x+y} + \frac{x^{2}}{y^{2}-x^{2}} $$.

1. **Rewrite the second fraction**:
   $$
   y^{2}-x^{2} = (y-x)(y+x)
   $$
   $$
   = \frac{x}{x+y} + \frac{x^{2}}{(y-x)(y+x)}
   $$
2. **Finding a common denominator**:
   $$
   = \frac{x(y-x)}{(x+y)(y-x)} + \frac{x^{2}(x+y)}{(y-x)(y+x)(x+y)}
   $$
3. **Combine the fractions**:
   $$
   = \frac{xy - x^{2} + x^{2}(x+y)}{(x+y)(y-x)}
   $$
   $$
   = \frac{xy + x^{3}}{(x+y)(y-x)}
   $$

**Final Result**: 
$$
\frac{xy}{y^{2}-x^{2}}
$$

These are the fully simplified results for each part of your questions.
1. $$ (x-2)(x^{2}+5x-1) = x^{3} + 3x^{2} - 11x + 2 $$ 2. $$ y^{2}(y-2) + x^{2}(2-y) = (y+x)(y-x)(y-2) $$ 3. - $$ \frac{x^{2}-4}{2x^{2}+5x+2} \div \frac{x^{3}-8}{6x+3} = \frac{3}{x^{2}+2x+4} $$ - $$ \frac{x}{x+y} + \frac{x^{2}}{y^{2}-x^{2}} = \frac{xy}{y^{2}-x^{2}} $$

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