1Factor $$\dfrac{x}{17}+\dfrac{y}{34}$$.
$$=\bigg( \dfrac{x}{17}\cdot\color{red}{\dfrac{2}{2}} \bigg)+\dfrac{y}{34} \\ \, \\ = \dfrac{x \times 2}{17 \times 2}+\dfrac{y}{34} \\ \, \\ = \dfrac{2x}{34}+\dfrac{y}{34} \\ \, \\= \dfrac{1}{34} (2x+y)$$

2Factor $$2(z+6) + 4yz+24y$$.
$$2(z+6) + 4yz+24y \\ \, \\ =2(z+6) + \underbrace{4yz+24y}_{\text{Factorize}} \\ \, \\ =2(z+6) + 4y(z+6) \\ \, \\ =(2+4y)(z+6)\\ \, \\ = 2(1+2y)(z+6)$$
3Factor $$\dfrac{9-6w}{15}+\dfrac{6w-8}{5}$$.
$$\blacktriangleright$$ Method 1: Using the factor $$\dfrac{1}{15}$$
$$\require{cancel} \dfrac{9-6w}{15}+\dfrac{6w-8}{5} \\ \, \\ =\dfrac{9-6w}{15}+\bigg(\dfrac{6w-8}{5}\cdot\color{red}{\dfrac{3}{3}} \bigg) \\ \, \\ = \dfrac{9-6w}{15}+\dfrac{3(6w-8)}{15} \\ \, \\ = \dfrac{9-6w}{15}+\underbrace{\dfrac{18w-24}{15}}_{\text{Expand}}\\ \, \\ = \dfrac{1}{15}(9-6w+18w-24) \\ \, \\ = \dfrac{1}{15}(12w-15) \\ \, \\ = \dfrac{1}{15}(3)(4w-5)\\ \, \\ = \dfrac{3}{15}(4w-5) \\ \, \\= \dfrac{\cancelto{1}{{\color{red}{3}}}}{\cancelto{5}{{\color{red}{15}}}}(4w-5) \\ \, \\= \dfrac{1}{5}(4w-5)$$
$$\blacktriangleright$$ Method 2: Using the factor $$\dfrac{1}{5}$$ by factoring the first expression first.
$$\require{cancel} \dfrac{9-6w}{15}+\dfrac{6w-8}{5} \\ \, \\ = \bigg(\dfrac{9-6w}{15}\bigg)\bigg(\dfrac{\frac{1}{3}}{\frac{1}{3}}\bigg)+\dfrac{6w-8}{5} \\ \, \\ = \dfrac{3-2w}{5}+\dfrac{6w-8}{5} \\ \, \\ = \dfrac{1}{5}(3-2w+6w-8) \\ \, \\ = \dfrac{1}{5}(4w-5)$$