Example : solve the equation \((x + 3) (x + 5) (x + 7) (x + 9) = 3465\)

Solution: Rewriting the expression we have, 

\([(x + 3)(x + 9)] [(x + 5)(x + 7)] =3465\)

\(\left( {{x^2} + {\rm{ }}12x + {\rm{ }}27} \right)\left( {{x^2} + {\rm{ }}12x + {\rm{ }}35} \right){\rm{ }} = {\rm{ }}3465\)

Now \({x^2} + {\rm{ }}12x + {\rm{ }}27{\rm{ }} = t\) \( \Rightarrow \) \(t(t + 8) = 3465\)

By simple observation

\(3465{\rm{ }} = \;63 \times 55{\rm{ }}or{\rm{ }}\left( {-{\rm{ }}55} \right) \times \left( {{\rm{ }}-{\rm{ }}63} \right)\)

Hence \(t = 55\) or - 63

Case 1: \(t = 55\), or \({x^2} + {\rm{ }}12x + {\rm{ }}27{\rm{ }} = {\rm{ }}55\)

or \({x^2} + {\rm{ }}12x-{\rm{ }}28{\rm{ }} = 0\)

\( \Rightarrow \)\((x + 14)(x - 2) = 0\) or \(x = -14, 2\)

When \(t = - 63\), equation becomes

\({x^2} + {\rm{ }}12x + {\rm{ }}27 = {\rm{ }}-63\)   or  \({x^2} + {\rm{ }}12x + {\rm{ }}90{\rm{ }} = {\rm{ }}0\)

or \(x = \frac{{ - 12 \pm \sqrt {144 - 360} }}{2} = \frac{{ - 12 \pm \sqrt { - 216} }}{2}\)

or \(x =  - 6 \pm i\sqrt {54} \)

Alternate solution: Real solutions of the equation can be found by factorizing the number 3465.

\(\left( {x + {\rm{ }}3} \right){\rm{ }}\left( {x + {\rm{ }}5} \right){\rm{ }}\left( {x + {\rm{ }}7} \right){\rm{ }}\left( {x + {\rm{ }}9} \right){\rm{ }} = {\rm{ }}5 \times 7 \times 9 \times 11\) or

\( = {\rm{ }}\left( {-{\rm{ }}5} \right) \times \left( {-{\rm{ }}7} \right) \times \left( {-{\rm{ }}9} \right) \times \left( {-11} \right)\)

\( \Rightarrow x + {\rm{ }}3{\rm{ }} = {\rm{ }}5\) or \(-{\rm{ }}11 \Rightarrow x = {\rm{ }}2,{\rm{ }}-{\rm{ }}14\)