Factorize

Problem:

$ 3x^2+14x+40 $

Solution:

The function cannot be factorized. We can do completing the square

$ \begin{eqnarray*} & & 3x^2+14x+40 &=& 3(x^2 + \frac{14}{3}x) + 40 \\ &=& 3(x^2 + 2\frac{7}{3}x) + 40 \\ &=& 3(x^2 + 2\frac{7}{3}x + \frac{49}{9} - \frac{49}{9}) + 40 \\ &=& 3(x^2 + 2\frac{7}{3}x + \frac{49}{9}) - \frac{49}{3} + 40 \\ &=& 3(x + \frac{7}{3})^2 - \frac{49}{3} + \frac{120}{3} \\ &=& 3(x + \frac{7}{3})^2 + \frac{71}{3} \\ \end{eqnarray*} $