Determine The HCF & LCM of the following: 1) a2-4; 8a2-14a-4; 12a2-21a-6 2) (x+2)2; 2x2-2y2; x2+2xy+y2 3) x2-16; 2x2-2x-24; x2-49 4) x2+3x-28; 2x+14; x2-49 5) 4a2-64; 2a-8b+2ab-8; b2-4b-5

For the first expression, $a^2-4$:
This is a difference of squares, $x^2-y^2 = (x-y)(x+y)$.

$$a^2-4 = (a-2)(a+2)$$

For the second expression, $8a^2-14a-4$:
First, factor out the common numerical factor, which is 2.

$$8a^2-14a-4 = 2(4a^2-7a-2)$$

Now, factor the quadratic $4a^2-7a-2$. We look for two numbers that multiply to $4 \times -2 = -8$ and add to $-7$. These numbers are $-8$ and 1.

$$4a^2-7a-2 = 4a^2-8a+a-2$$

Factor by grouping:

$$4a(a-2)+1(a-2) = (4a+1)(a-2)$$

So, the second expression is:

$$8a^2-14a-4 = 2(4a+1)(a-2)$$

For the third expression, $12a^2-21a-6$:
First, factor out the common numerical factor, which is 3.

$$12a^2-21a-6 = 3(4a^2-7a-2)$$

As determined above, $4a^2-7a-2 = (4a+1)(a-2)$.
So, the third expression is:

$$12a^2-21a-6 = 3(4a+1)(a-2)$$

The factored expressions are:

1. $(a-2)(a+2)$
2. $2(4a+1)(a-2)$
3. $3(4a+1)(a-2)$

To find the HCF, we look for factors that are common to all three expressions.

• The factor $(a-2)$ is present in expressions 1 and 2, but not in expression 3.
• The factor $(a+2)$ is present only in expression 1.
• The factor $(4a+1)$ is present in expressions 2 and 3, but not in expression 1.
• There are no common numerical factors across all three expressions (1, 2, and 3). Since there are no common factors other than 1 across all three expressions, the HCF is 1.

$$\text{HCF} = 1$$

The LCM is the product of all unique factors (common and non-common) raised to the highest power they appear in any of the expressions.
The unique factors are:

• Numerical factors: 2 (from expression 2) and 3 (from expression 3). The highest power of 2 is $2^1$. The highest power of 3 is $3^1$. So, $2 \times 3 = 6$.
• Algebraic factors:
  • $(a-2)$: Appears in expressions 1 and 2. The highest power is 1.
  • $(a+2)$: Appears in expression 1. The highest power is 1.
  • $(4a+1)$: Appears in expressions 2 and 3. The highest power is 1.

Multiply all these unique factors with their highest powers:

$$\text{LCM} = 6(a-2)(a+2)(4a+1)$$

The HCF and LCM are:
HCF: $\boxed{1}$
LCM: $\boxed{6(a-2)(a+2)(4a+1)}$