Let’s factorise each expression by taking out the common factors.

Solutions

a) $2x^2 - 32$

• Factor out 2:
  $2(x^2 - 16)$
• Further factorize $x^2 - 16$ as a difference of squares:
  $2(x + 4)(x - 4)$

b) $3x^2 - 108$

• Factor out 3:
  $3(x^2 - 36)$
• Further factorize $x^2 - 36$:
  $3(x + 6)(x - 6)$

c) $4a^2 - 100$

• Factor out 4:
  $4(a^2 - 25)$
• Further factorize $a^2 - 25$:
  $4(a + 5)(a - 5)$

d) $5y^2 - 20$

• Factor out 5:
  $5(y^2 - 4)$
• Further factorize $y^2 - 4$:
  $5(y + 2)(y - 2)$

e) $24a^2 - 6b^2$

• Factor out 6:
  $6(4a^2 - b^2)$
• Further factorize $4a^2 - b^2$:
  $6(2a + b)(2a - b)$

f) $3x^2 - 27y^2$

• Factor out 3:
  $3(x^2 - 9y^2)$
• Further factorize $x^2 - 9y^2$:
  $3(x + 3y)(x - 3y)$

g) $8y^2 - 128$

• Factor out 8:
  $8(y^2 - 16)$
• Further factorize $y^2 - 16$:
  $8(y + 4)(y - 4)$

h) $80p^2 - 5q^2$

• Factor out 5:
  $5(16p^2 - q^2)$
• Further factorize $16p^2 - q^2$:
  $5(4p + q)(4p - q)$

i) $4x^2 - 64$

• Factor out 4:
  $4(x^2 - 16)$
• Further factorize $x^2 - 16$:
  $4(x + 4)(x - 4)$

j) $3x^2 - 3$

• Factor out 3:
  $3(x^2 - 1)$
• Further factorize $x^2 - 1$:
  $3(x + 1)(x - 1)$

k) $72p^2 - 2$

• Factor out 2:
  $2(36p^2 - 1)$
• Further factorize $36p^2 - 1$:
  $2(6p + 1)(6p - 1)$

l) $2 - 18x^2$

• Factor out 2:
  $2(1 - 9x^2)$
• Further factorize $1 - 9x^2$:
  $2(1 + 3x)(1 - 3x)$

m) $8a^2 - 18m^2$

• Factor out 2:
  $2(4a^2 - 9m^2)$
• Further factorize $4a^2 - 9m^2$:
  $2(2a + 3m)(2a - 3m)$

n) $125 - 20a^2$

• Factor out 5:
  $5(25 - 4a^2)$
• Further factorize $25 - 4a^2$:
  $5(5 + 2a)(5 - 2a)$

o) $200x^2 - 18y^2$

• Factor out 2:
  $2(100x^2 - 9y^2)$
• Further factorize $100x^2 - 9y^2$:
  $2(10x + 3y)(10x - 3y)$

p) $98m^2 - 8n^2$

• Factor out 2:
  $2(49m^2 - 4n^2)$
• Further factorize $49m^2 - 4n^2$:
  $2(7m + 2n)(7m - 2n)$

Would you like further clarification on any of these steps?

Related Questions:

1. How do you determine if a quadratic expression can be factored further?
2. What is the difference between factoring out a common factor and factoring a difference of squares?
3. How do you identify the greatest common factor in polynomials with multiple terms?
4. What is the procedure for factoring a trinomial versus a binomial?
5. How can factorization be applied in solving algebraic equations?

Tip:

Always check if the terms share a greatest common factor before attempting to use other factoring methods. This step can simplify the process significantly.