a)

$2y\left(y^3+6y-9\right)=2y\cdot y^3+2y\cdot 6y+2y\cdot \left(-9\right)=2y^4+12y^2-18y$

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b)

$\left(\frac{1}{2}{b}^3-\frac{1}{6}{b}^2+4\right)\cdot \left(-6b^3\right)=\frac{1}{2}{b}^3\cdot \left(-6b^3\right)-\frac{1}{6}{b}^2\cdot \left(-6b^3\right)+4\cdot \left(-6b^3\right)=-3b^6+b^5-24b^3$

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c)

$\left(14y^5+21y^2-5y\right):\left(-7\right)=14y^5:\left(-7\right)+21y^2:\left(-7\right)-5y:\left(-7\right)=-2y^5-3y^2+\frac{5}{7}y$

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d)

$\frac{25y{z}^2-7y^{2}z+10z}{5}=\frac{25y{z}^2}{5}-\frac{7y^{2}z}{5}+\frac{10z}{5}=5y{z}^2-\frac{7}{5}{y}^{2}z+2z=5y{z}^2-1\frac{2}{5}{y}^{2}z+2z$