$a)y=-\frac{1}{3}\left(x+3\right)\left(x-9\right)=$

$=-\frac{1}{3}\left(x\left(x-9\right)+3\left(x-9\right)\right)=$

$=-\frac{1}{3}\left(x^2-9x+3x-27\right)=$

$=-\frac{1}{3}\left(x^2-6x-27\right)=$

$=-\frac{1}{3}{x}^2+2x+9$

 

$b)y=4\left(x-\frac{1}{2}\right)\left(x-\frac{1}{3}\right)=$

$=4\left(x\left(x-\frac{1}{3}\right)-\frac{1}{2}\left(x-\frac{1}{3}\right)\right)=$

$=4\left(x^2-\frac{1}{3}x-\frac{1}{2}x+\frac{1}{6}\right)=$

$=4\left(x^2-\frac{2}{6}x-\frac{3}{6}x+\frac{1}{6}\right)=$

$=4\left(x^2-\frac{5}{6}x+\frac{1}{6}\right)=$

$=4x^2-\frac{20}{6}x+\frac{4}{6}=$

$=4x^2-\frac{10}{3}x+\frac{2}{3}$

 

 

$c)y=\frac{1}{6}\left(x+2\right)x=$

$=\frac{1}{6}\left(x^2+2x\right)=$

$=\frac{1}{6}{x}^2+\frac{1}{3}x$