#"before subtracting the fractions we require them to "#
#"have a "color(blue)"common denominator"#
#"factorising the denominators of both fractions"#
#•color(white)(x)4y^2+y=y(4y+1)#
#•color(white)(x)4y^2-23y-6#
#"the factors of the product - 24 which sum to - 23 are"#
#"are - 24 and + 1"#
#rArr4y^2-24y+y-6#
#=color(red)(4y)(y-6)color(red)(+1)(y-6)larrcolor(blue)"factor by grouping"#
#=(y-6)(color(red)(4y+1))larrcolor(blue)"common factor "(y-6)#
#rArr(2y)/(y(4y+1))-(3+y)/((4y+1)(y-6))#
#"the common denominator is "y(4y+1)(y-6)#
#"multiply numerator/denominator of "#
#(2y)/(y(4y+1))" by "(y-6)#
#"multiply numerator/denominator of"#
#(3+y)/((4y+1)(y-6))" by "y#
#rArr(2y(y-6))/(y(4y+1)(y-6))-(y(3+y))/(y(4y+1)(y-6))#
#"we now have a common denominator so can subtract"#
#"the numerators leaving the denominator"#
#=(2y^2-12y-3y-y^2)/(y(4y+1)(y-6))#
#=(y^2-15y)/(y(4y+1)(y-6))#
#=(cancel(y)(y-15))/(cancel(y)(4y+1)(y-6))=(y-15)/((4y+1)(y-6))#
#"with restrictions "y!=-1/4,y!=6#
