Factorize each of the following :
i)\(8a^3 + b^3 + 12a^2b + 6ab^2\)
ii)\(8a^3 - b^3 - 12a^2b + 6ab^2\)
iii)\(27 - 125a^3 - 135a + 225a^2\)
iv)\(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
v)\(27p^3 - \frac{1}{216} - \frac{9p^2}{2} + \frac{p}{4} \)

i)\(8a^3 + b^3 + 12a^2b + 6ab^2 \)
\(= (2a)^3 + b^3 +3(2a)(b)(2a + b))\)
\([\because (a + b)^3 = a^3 + b^3 + 3ab(a + b)]\)
\(= (2a)^3 + b^3\)


ii)\(8a^3 - b^3 - 12a^2b + 6ab^2 \)
\(= (2a)^3 - b^3 -3(2a)(b)(2a - b)\)
\([\because (a - b)^3 = a^3 - b^3 - 3ab(a - b)] \)
\(= (2a)^3 - b^3\)


iii)\(27 - 125a^3 - 135a + 225a^2 \)
\(= (3)^3 - (5a)^3 -3(3)(5a)(3 - 5a)\)
\([\because (a - b)^3 = a^3 - b^3 - 3ab(a - b)]\)
\(= (3)^3 - 5a^3\)


iv)\(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
\( = (4a)^3 - (3b)^3 -3(4a)(4b)(4a - 3b))\)
\([\because (a - b)^3 = a^3 - b^3 - 3ab(a - b)]\)
\(= (4a)^3 - (3b)^3\)


v)\((27p^3 - \frac{1}{216} - \frac{9p^2}{2} + \frac{p}{4} \)
\( = (3p)^3 - (\frac{1}{6} )^3 -3(3p)(\frac{1}{6})(3p - 1/6))\)
\([\because (a - b)^3 = a^3 - b^3 - 3ab(a - b)] \)
\(= (3p)^3 - (\frac{1}{6})^3\)