# Divide ((6x^2+36xy+54)/(x^2+5xy+4y^2))÷((7x+21y)/(x+y))

6×2+36xy+54×2+5xy+4y2÷7x+21yx+y
To divide by a fraction, multiply by its reciprocal.
6×2+36xy+54×2+5xy+4y2⋅x+y7x+21y
Factor 6 out of 6×2+36xy+54.
Factor 6 out of 6×2.
6(x2)+36xy+54×2+5xy+4y2⋅x+y7x+21y
Factor 6 out of 36xy.
6(x2)+6(6xy)+54×2+5xy+4y2⋅x+y7x+21y
Factor 6 out of 54.
6×2+6(6xy)+6⋅9×2+5xy+4y2⋅x+y7x+21y
Factor 6 out of 6×2+6(6xy).
6(x2+6xy)+6⋅9×2+5xy+4y2⋅x+y7x+21y
Factor 6 out of 6(x2+6xy)+6⋅9.
6(x2+6xy+9)x2+5xy+4y2⋅x+y7x+21y
6(x2+6xy+9)x2+5xy+4y2⋅x+y7x+21y
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=1⋅4=4 and whose sum is b=5.
Reorder terms.
6(x2+6xy+9)x2+4y2+5xy⋅x+y7x+21y
Reorder 4y2 and 5xy.
6(x2+6xy+9)x2+5xy+4y2⋅x+y7x+21y
Factor 5 out of 5xy.
6(x2+6xy+9)x2+5(xy)+4y2⋅x+y7x+21y
Rewrite 5 as 1 plus 4
6(x2+6xy+9)x2+(1+4)(xy)+4y2⋅x+y7x+21y
Apply the distributive property.
6(x2+6xy+9)x2+1(xy)+4(xy)+4y2⋅x+y7x+21y
Multiply xy by 1.
6(x2+6xy+9)x2+xy+4(xy)+4y2⋅x+y7x+21y
Move parentheses.
6(x2+6xy+9)x2+xy+4xy+4y2⋅x+y7x+21y
6(x2+6xy+9)x2+xy+4xy+4y2⋅x+y7x+21y
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
6(x2+6xy+9)(x2+xy)+4xy+4y2⋅x+y7x+21y
Factor out the greatest common factor (GCF) from each group.
6(x2+6xy+9)x(x+y)+4y(x+y)⋅x+y7x+21y
6(x2+6xy+9)x(x+y)+4y(x+y)⋅x+y7x+21y
Factor the polynomial by factoring out the greatest common factor, x+y.
6(x2+6xy+9)(x+y)(x+4y)⋅x+y7x+21y
6(x2+6xy+9)(x+y)(x+4y)⋅x+y7x+21y
Cancel the common factor of x+y.
Cancel the common factor.
6(x2+6xy+9)(x+y)(x+4y)⋅x+y7x+21y
Rewrite the expression.
6(x2+6xy+9)x+4y⋅17x+21y
6(x2+6xy+9)x+4y⋅17x+21y
Multiply 6(x2+6xy+9)x+4y and 17x+21y.
6(x2+6xy+9)(x+4y)(7x+21y)
Factor 7 out of 7x+21y.
Factor 7 out of 7x.
6(x2+6xy+9)(x+4y)(7(x)+21y)
Factor 7 out of 21y.
6(x2+6xy+9)(x+4y)(7(x)+7(3y))
Factor 7 out of 7(x)+7(3y).
6(x2+6xy+9)(x+4y)(7(x+3y))
6(x2+6xy+9)(x+4y)⋅7(x+3y)
Move 7 to the left of x+4y.
6(x2+6xy+9)7(x+4y)(x+3y)
Apply the distributive property.
6×2+6(6xy)+6⋅97(x+4y)(x+3y)
Simplify.
Multiply 6 by 6.
6×2+36(xy)+6⋅97(x+4y)(x+3y)
Multiply 6 by 9.
6×2+36(xy)+547(x+4y)(x+3y)
6×2+36xy+547(x+4y)(x+3y)
Split the fraction 6×2+36xy+547(x+4y)(x+3y) into two fractions.
6×2+36xy7(x+4y)(x+3y)+547(x+4y)(x+3y)
Split the fraction 6×2+36xy7(x+4y)(x+3y) into two fractions.
6×27(x+4y)(x+3y)+36xy7(x+4y)(x+3y)+547(x+4y)(x+3y)
Divide ((6x^2+36xy+54)/(x^2+5xy+4y^2))÷((7x+21y)/(x+y))