Problems from Notable products

Using notable products develop the following algebraic expressions:

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1) $(x + y + 1)^{2} = (x + (y + 1))^{2} = x^{2} + 2 \cdot x \cdot (y + 1) + (y + 1)^{2} =$

$= x^{2} + 2 x y + 2 x + (y^{2} + 2 y + 1) = x^{2} + y^{2} + 2 x y + 2 x + 2 y + 1$

2) $(x + 1) \cdot (x - 1) \cdot (x - 1) \cdot (x + 1) =$

$= ((x + 1) \cdot (x - 1)) \cdot ((x - 1) \cdot (x + 1)) =$

$= (x^{2} - 1) \cdot (x^{2} - 1) = (x^{2} - 1)^{2} = x^{4} - 2 x^{2} + 1$

3) $(x^{2} - 2)^{3} = (x^{2})^{3} - 3 \cdot (x^{2})^{2} \cdot 2 + 3 \cdot x^{2} \cdot 2^{2} - 2^{3} = x^{6} - 6 x^{4} + 12 x^{2} - 8$

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