Find the LCM of $x^{2} + 4 x + 4, x^{2} + 5 x + 6$

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Solution

We know that $L C M$ is the least common multiple.

Factorise $x^{2} + 4 x + 4$ as follows:

$x^{2} + 4 x + 4 = x^{2} + 2 x + 2 x + 4 = x (x + 2) + 2 (x + 2) = (x + 2) (x + 2) = (x + 2)^{2}$

Now, factorise $x^{2} + 5 x + 6$ as follows:

$x^{2} + 5 x + 6 = x^{2} + 2 x + 3 x + 6 = x (x + 2) + 3 (x + 2) = (x + 3) (x + 2)$

Therefore, the least common multiple between the polynomials $x^{2} + 4 x + 4$ and $x^{2} + 5 x + 6$ is:

$L C M = (x + 2)^{2} (x + 3)$

Hence, the $L C M$ is $(x + 2)^{2} (x + 3)$ .

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