Question 186835
{{{y/(4y+8) - 1/(y^2+2y)}}} Start with the given expression.



{{{y/(4(y+2)) - 1/(y^2+2y)}}} Factor {{{4y+8}}} to get {{{4(y+2)}}}



{{{y/(4(y+2)) - 1/(y(y+2))}}} Factor {{{y^2+2y}}} to get {{{y(y+2)}}}



Now looking at the denominators, we have the factors:


{{{4}}}, {{{y+2}}}, {{{y}}} and {{{y+2}}}



Now simply find the LCM of {{{4}}}, {{{y+2}}}, {{{y}}} and {{{y+2}}}


So find the unique factors (and the most frequent unique factors) and multiply them to get: {{{4y(y+2)}}}



So the LCD is {{{4y(y+2)}}}. If this does not make any sense, then let me know as it is critical to grasp this concept. 



Now the goal is to get EVERY denominator equal to the LCD. 




{{{(y*highlight(y))/(4*highlight(y)(y+2)) - 1/(y(y+2))}}} Multiply both the numerator and denominator of the first fraction by {{{y}}} (to get this denominator equal to the LCD)




{{{(y^2)/(4y(y+2)) - 1/(y(y+2))}}} Multiply.



{{{(y^2)/(4y(y+2)) - (1*highlight(4))/(highlight(4)*y(y+2))}}} Multiply both the numerator and denominator by of the second fraction {{{4}}} (to get this denominator equal to the LCD)



{{{(y^2)/(4y(y+2)) - (4)/(4y(y+2))}}} Multiply



{{{(y^2-4)/(4y(y+2))}}} Combine the fractions (this is now possible since EVERY fraction has an equal denominator).



{{{((y+2)(y-2))/(4y(y+2))}}} Factor the numerator



{{{(highlight((y+2))(y-2))/(4y*highlight((y+2)))}}} Highlight the common terms.



{{{(cross((y+2))(y-2))/(4y*cross((y+2)))}}} Cancel out the common terms.



{{{(y-2)/(4y)}}} Simplify




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Answer:



So {{{y/(4y+8) - 1/(y^2+2y)=(y-2)/(4y)}}} where {{{y<>-2}}} or {{{y<>0}}}