Simplify:
$(2 a + b)^{3} - (2 a - b)^{3}$

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Solution

${(2 a + b)}^{3} - {(2 a - b)}^{3}$

Using identity:-

${(x + y)}^{3} = x^{3} + y^{3} + 3 x^{2} y + 3 x y^{2}$

${(x - y)}^{3} = x^{3} - y^{3} - 3 x^{2} y + 3 x y^{2}$

Therefore,

${(2 a + b)}^{3} - {(2 a - b)}^{3}$

$= [{(2 a)}^{3} + b^{3} + 3 {(2 a)}^{2} b + 2 (2 a) b^{2}] - [{(2 a)}^{3} - b^{3} - 3 {(2 a)}^{2} b + 2 (2 a) b^{2}]$

$= [8 a^{3} + b^{3} + 12 a^{2} b + 3 a b^{2}] - [8 a^{3} - b^{3} - 12 a^{2} b + 3 a b^{2}]$

$= 8 a^{3} + b^{3} + 12 a^{2} b + 3 a b^{2} - 8 a^{3} + b^{3} + 12 a^{2} b - 3 a b^{2}$

$= b^{3} + 12 a^{2} b + b^{3} + 12 a^{2} b$

$= 2 b^{3} + 24 a^{2} b$

Hence ${(2 a + b)}^{3} - {(2 a - b)}^{3} = 2 b^{3} + 24 a^{2} b$ .

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Factorise :

(a + b)³ - 8

My attempt

= (a + b)³- (2)³

= (a + b - 2) {(a + b)²+ (2)² + 2(a + b)}

What should I do not?

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