$a)2a^{2}$-$bc^{2}$-$a^{2}b+2c^{2}$=$a^{2}(2-b)+c^{2}(2-b)$=$(a^{2}+c^{2})(2-b)$
$b)6x+3-(2x-5)(2x+1)=3(2x+1)-(2x-5)(2x+1)=(-2x+8)(2x+1)=-2(x-4)(x-1)$
$c)ab^{3}-a^{2}b^{2}c^{2}+ab^{2}c^{3}-a^{2}bc^{3}=ab(b^{2}-abc^{2}+bc^{3}-ac^{3})$
$d)ax^{2}+cx^{2}-ay-ay^{2}-cy-cy^{2}=a(x^{2}-y-y^{2})+c(x^{2}-y-y^{2})=(c+a)(x^{2}-y-y^{2})$
$e)5xy^{3}-2xyz-15y^{2}-6z=(5xy^{3}+15y^{2})-(2xyz+6z)=5y^{2}(xy+3)-2z(xy+3)$
$=(5y^{2}-2z)(xy+3)$