$\text{a)}y=-\frac{1}{3}\left(x+3\right)\left(x-9\right)=-\frac{1}{3}\left(x^2-9x+3x-27\right)=-\frac{1}{3}\left(x^2-6x-27\right)=-\frac{1}{3}{x}^2+2x+9$  

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$\text{b)}y=4\left(x-\frac{1}{2}\right)\left(x-\frac{1}{3}\right)=4\left(x^2-\frac{1}{3}x-\frac{1}{2}x+\frac{1}{6}\right)=4\left(x^2-\frac{2}{6}x-\frac{3}{6}x+\frac{1}{6}\right)=4\left(x^2-\frac{5}{6}x+\frac{1}{6}\right)=4x^2-\frac{10}{3}x+\frac{2}{3}$  

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$\text{c)}y=\frac{1}{6}x\left(x+2\right)=\frac{1}{6}{x}^2+\frac{1}{3}x$