Factoring By Substitution

Factoring by substitution means that substituting the value(s) of variables by a common and single variable, by which the given equation yields the simpler form of the given expression which is easy to factor.

Decompose the following expression into a product of factors:

\((x^3 + 2x^2 + 3x + 5)(x^3 + 2x^2 + 3x + 15) +21.\)

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First, let's name the first expression as \(t\):

\[x^3 + 2x^2 + 3x + 5 = t.\]

Then the main expression will become

\[\begin{align} E(t) &= t(t + 10) + 21 \\ &=t^2 + 10t + 21 \\ &=(t + 3)(t + 7). \end{align}\]

Now, we return to our main expression, and the decomposed expression in a product of factors is \[\begin{align} E(x^3 + 2x^2 + 3x + 5) &= \left[\left(x^3 + 2x^2 + 3x +5\right)+3\right]\left[\left(x^3 + 2x^2 + 3x + 5\right)+7\right]\\ &= \left(x^3 + 2x^2 + 3x +8\right)\left(x^3 + 2x^2 + 3x + 12\right). \ _\square \end{align}\]