Answer
$\left(-\displaystyle \frac{3}{2}, -\displaystyle \frac{7}{4}\right)$.
Work Step by Step
$ f(x)=a(x-h)^{2}+k^{2},\qquad$ vertex: (h,k)
(solved as in example 4)
$(A\pm B)^{2}=A^{2}\pm 2AB+B^{2}$
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$y=3x^{2}+9x+5$
$=3(x^{2}+3x)+5$
$=3(x^{2}+2\displaystyle \cdot\frac{3}{2}\cdot x+\frac{9}{4}-\frac{9}{4})+5$
$=3(x^{2}+3x+\displaystyle \frac{9}{4})+5-3(\frac{9}{4})$
$=3(x+\displaystyle \frac{3}{2})^{2}-\frac{20-27}{4}$
$=3(x+\displaystyle \frac{3}{2})^{2}-\frac{7}{4}$
Vertex: $(-\displaystyle \frac{3}{2}, -\displaystyle \frac{7}{4})$.
