Find the product of the binomials $x + 2, x + 3$ and $x + 4$ .

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Solution

We use the identity
$(x + a) (x + b) (x + c) = x^{3} + x^{2} (a + b + c) + x (a b + b c + c a) + a b c$
Take $a = 2, b = 3$ and $c = 4$ . You obtain
$a + b + c = 2 + 3 + 4 = 9$
$a b + b c + c a = (2 \times 3) + (3 \times 4) + (4 \times 2) = 6 + 12 + 18 = 26$ ,
$a b c = 2 \times 3 \times 4 = 24$ .
Hence
$(x + 2) (x + 3) (x + 4) = x^{3} + 9 x^{2} + 26 x + 24$

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