**Question 1: Divide x**^{2} – 5x + 6 by (x – 3)

^{2}– 5x + 6 by (x – 3)

**Solution:**

(x2 – 5x + 6)/(x-3)

Factorise the numerator and then divide it by (x-3):

x

^{2}– 5x + 6= x

^{2}– 3x – 2x + 6= (x

^{2}– 3x – 2x + 6)/(x – 3)= (x(x – 3) – 2(x – 3))/(x – 3)

= ((x – 3)(x – 2))/(x – 3)

= (x – 2)

Therefore, the answer is (x-2).

**Question 2: Divide ax**^{2} – ay^{2} by (ax + ay)

^{2}– ay

^{2}by (ax + ay)

**Solution:**

(ax

^{2}– ay^{2})/(ax + ay)= a(x

^{2}– y^{2})/(ax + ay)= a(x – y)(x + y)/a(x + y)

= x – y

Therefore, the answer is (x – y).

**Question 3: Divide (x**^{4} – y^{4)} by (x^{2} – y^{2})

^{4}– y

^{4)}by (x

^{2}– y

^{2})

**Solution:**

(x

^{4}– y^{4})/(x^{2}– y^{2})= ((x

^{2})^{2}– (y^{2})^{2})/(x^{2}– y^{2})= ((x

^{2}– y^{2}) (x^{2}+ y^{2})) / (x^{2}– y^{2})= x

^{2}+ y^{2}Therefore, the answer is (x

^{2 }+ y^{2}).

**Question 4: Divide (acx**^{2} + (bc + ad)x + bd) by (ax + b)

^{2}+ (bc + ad)x + bd) by (ax + b)

**Solution:**

(acx

^{2}+ (bc + ad)x + bd) / (ax + b)= (acx

^{2}+ bcx + adx + bd) / (ax + b)= (cx(ax + b) + d(ax + b)) / (ax + b)

= (ax + b)(cx + d) / (ax + b)

= cx + d

Therefore, the answer is (cx + d).

**Question 5: Divide (a**^{2} + 2ab + b^{2}) – (a^{2} + 2ac + c^{2}) by (2a + b + c)

^{2}+ 2ab + b

^{2}) – (a

^{2}+ 2ac + c

^{2}) by (2a + b + c)

**Solution:**

((a

^{2}+ 2ab + b^{2}) – (a^{2}+ 2ac + c^{2})) / (2a + b + c)= ((a + b)

^{2}– (a + c)^{2}) / (2a + b + c)= ((a + b + a + c)(a + b – a – c)) / (2a + b + c)

= (2a + b + c)(b – c) / (2a + b + c)

= b – c

Therefore, the answer is (b – c).

**Question 6: Divide ((1/4)x**^{2 }– (1/2)x – 12) by ((1/2)x – 4)

^{2 }– (1/2)x – 12) by ((1/2)x – 4)

**Solution:**

(1/4)x

^{2}– (1/2)x – 12= (1/4)(x

^{2}– 2x – 48)Now factorize it:

= (1/4)(x

^{2}– 8x + 6x – 48)= (1/4)(x(x – 8) + 6(x – 8))

= (1/4)(x – 8)(x + 6)

Now divide it by (1/2)x – 4:

= (1/4)(x – 8)(x + 6) / ((1/2)x – 4)

= (1/4)(x – 8)(x + 6) / (1/2)(x – 8)

= (1/4)(2/1)(x + 6)

= (1/2)x + 3

Therefore, the answer is (1/2)x + 3.