Question

simplify:
5x/(x^2-x-6) - 4/(x^2+4x+4)

Answer 1

Factor
(5x)/((x+2)(x-3))-(4)/(x^(2)+4x+4)

Factor
(5x)/((x+2)(x-3))-(4)/((x+2)(x+2))

add 1 to 1 to get 2.
(5x)/((x+2)(x-3))-(4)/((x+2)^(2))

Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of (x-3)(x+2)^(2).


The (5x)/((x+2)(x-3)) expression needs to be multiplied by ((x+2))/((x+2)) to make the denominator (x-3)(x+2)^(2). The -(4)/((x+2)^(2)) expression needs to be multiplied by ((x-3))/((x-3)) to make the denominator (x-3)(x+2)^(2).
(5x)/((x+2)(x-3))*(x+2)/(x+2)-(4)/((x+2)^(2))*(x-3)/(x-3)

Multiply the expression by a factor of 1 to create the least common denominator of (x-3)(x+2)^(2)
(5x(x+2))/((x-3)(x+2)^(2))-(4(x-3))/((x-3)(x+2)^(2))

The numerators of expressions that have equal denominators can be combined. In this case, (5x(x+2))/((x-3)(x+2)^(2)) and -((4(x-3)))/((x-3)(x+2)^(2)) have the same denominator of (x-3)(x+2)^(2), so the numerators can be combined.
(5x(x+2)-(4(x-3)))/((x-3)(x+2)^(2))

combine all similar expressions in the polynomial
(5x^(2)+6x+12)/((x-3)(x+2)^(2))

Answer 2

Thank you.