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We need to understand what we mean by LCM.

LCM of numbers is the smallest positive number that is multiple of two or more numbers.

Now we need to find the LCM of three numbers, which are $ {{\text{m}}^{2}}-9m-22 $ , $ {{m}^{2}}-8m-33 $ , $ {{m}^{2}}+5m+6 $ .

We need to factorize all the terms.

Let's start with the first term $ {{\text{m}}^{2}}-9m-22 $ .

$ \begin{align}

& {{\text{m}}^{2}}-9m-22={{\text{m}}^{2}}-11m+2m-22 \\

& =m\left( m-11 \right)+2\left( m-11 \right)

\end{align} $

Taking $ \left( m-11 \right) $ common we get,

$ {{\text{m}}^{2}}-9m-22=\left( m+2 \right)\left( m-11 \right)....................(i) $

Now, let's factorize the second number $ {{m}^{2}}-8m-33 $ .

$ \begin{align}

& {{\text{m}}^{2}}-8m-33={{\text{m}}^{2}}-11m+3m-33 \\

& =m\left( m-11 \right)+3\left( m-11 \right)

\end{align} $

Taking $ \left( m-11 \right) $ common we get,

$ {{\text{m}}^{2}}-8m-33=\left( m+3 \right)\left( m-11 \right)....................(ii) $

Now, let's factorize the third number $ {{m}^{2}}+5m+6 $ .

$ \begin{align}

& {{\text{m}}^{2}}+5m+6={{\text{m}}^{2}}+2m+3m+6 \\

& =m\left( m+2 \right)+3\left( m+2 \right)

\end{align} $

Taking $ \left( m+2 \right) $ common we get,

$ {{\text{m}}^{2}}+5m+6=\left( m+2 \right)\left( m+3 \right)....................(ii) $

Now, to calculate LCM we need to multiply all the values at least once.

By doing that we get,

$ L.C.M.=\left( m+2 \right)\left( m+3 \right)\left( m-11 \right) $

Hence, the L.C.M of these three terms is $ \left( m+2 \right)\left( m+3 \right)\left( m-11 \right) $ .

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