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$${(a + b)^2} = {a^2} + 2ab + {b^2}$$
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$${(a – b)^2} = {a^2} – 2ab + {b^2}$$
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$${(a + b)^2} = {(a – b)^2} + 4ab$$
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$${(a – b)^2} = {(a + b)^2} – 4ab$$
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$${(a + b)^2} + {(a – b)^2} = 2{a^2} + 2{b^2}$$
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$${(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac$$
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$${(a + b + c + \cdots )^2} = {a^2} + {b^2} + {c^2} + \cdots + 2(ab + ac + bc + \cdots )$$
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$${(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} = {a^3} + {b^3} + 3ab(a + b)$$
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$${(a – b)^3} = {a^3} – 3{a^2}b + 3a{b^2} – {b^3} = {a^3} – {b^3} – 3ab(a – b)$$
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$$(a + b)(a – b) = {a^2} – {b^2}$$
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$${a^3} – {b^3} = (a – b)({a^2} + ab + {b^2})$$
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$${a^3} + {b^3} = (a + b)({a^2} – ab + {b^2})$$
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$$(a + b)(a + c) = {a^2} + (b + c)a + bc$$
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$$(x + b)(x + c) = {x^2} + (b + c)x + bc$$
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$$(a + b + c)({a^2} + {b^2} + {c^2} – ac – bc – ca) = {a^3} + {b^3} + {c^3} – 3abc$$
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$${a^n} – {b^n} = (a – b)({a^{n – 1}} + {a^{n – 2}}b + {a^{n – 3}}{b^2} + \cdots + {b^{n – 1}})$$ if $$n$$ is odd.
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$${a^n} – {b^n} = (a + b)({a^{n – 1}} – {a^{n – 2}}b + {a^{n – 3}}{b^2} – \cdots – {b^{n – 1}})$$ if $$n$$ is even.
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$${a^n} + {b^n} = (a + b)({a^{n – 1}} – {a^{n – 2}}b + {a^{n – 3}}{b^2} – \cdots – {b^{n – 1}})$$ if $$n$$ is odd.
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$$(x + a)(x + b)(x + c) = {x^3} + (a + b + c){x^2} + (ab + bc + ac)x + abc$$