Question 1: Factor the following polynomial: \( \left( {{x^2} + 2x + 5} \right)\left( {{x^2} + 2x + 3} \right) – 15 \)

\(\begin{array}{l} \left( {{x^2} + 2x + 5} \right)\left( {{x^2} + 2x + 3} \right) – 15\\ = \ left[ {\left( {{x^2} + 2x + 4} \right) + 1} \right]\left[ {\left( {{x^2} + 2x + 4} \right) – 1} \right] – 15\\ = {\left( {{x^2} + 2x + 4} \right)^2} – {1^2} – 15 = {\left( {{x^2} + 2x + 4} \right)^2} – {4^2}\\ = \left( {{x^2} + 2x + 4 – 4} \right)\left( {{x^2} + 2x + 4 + 4} \right)\\ = \left( {{x^2} + 2x} \right)\left( {{x^2} + 2x + 8} \right) = x\left( {x + 2} \right )\left( {{x^2} + 2x + 4} \right) \end{array}\)