Please show me how to simplify this expression and solve the equation:

Question

(v^2 + 6v+8)/(v^2 + v - 12) / (v+2)/(2v-6)

Thank you very much.

Answer 1

What was posted does not represent an equation. Please check before posting.

If I assume the second / is an equal sign, then the equation reads:
(v^2 + 6v+8)/(v^2 + v - 12) = (v+2)/(2v-6)

To solve this equation, we start by factoring the quadratic expression into:
(v²+6v+8)=(v+2)(v+4)
and
(v²+v-12)=(v+4)(v-3)
also
(2v-6)=2(v-3)

Putting this altogether, equation becomes:
(v+2)(v+4)/(v+4)(v-3)=(1/2)(v+2)/(v-3)

If (v+2)≠0, (v+4)≠0, (v-3)≠0, then we can cancel common factors on left and right sides, we end up with
1=1/2
which means that our assumption is wrong.
(v+4)≠0 because else the left-hand-side will be indeterminate.
(v-3)≠0 because else both sides will be infinite.
The last possible option is (v+2)=0, which makes both sides equal (0=0).
By elimination, the only possible answer is v+2=0, or v=-2.

Answer 2

The way it was typed and following the order of operation, we have:

(v^2 + 6v+8)/(v^2 + v - 12) / (v+2)/(2v-6)
= (v+2)(v+4)/( (v+4)(v-3)(v+2)(2)(v-3) )
= 1/(2(v-3)^2)

If you meant:
(v^2 + 6v+8)/[ (v^2 + v - 12) / ((v+2)/(2v-6)) ]
then we would have
(v+2)(v+4)/[ (v+4)(v-3)/((v+2)/(2)(v-3)) ]
= (v+2)(v+4) * (2(v+2)(v-3)/( (v+4)(v-3) )
= 2(v+2)^2

e.g. 24/3/4
= (24/3) / 4 = 8/4 = 2 , my first interpretation
matching your typed question
http://www.wolframalpha.com/input/?i=calculate+24%C3%B73%C3%B74

or

= 24 / (3/4)
= 24 * (4/3) = 32 , the second interpretation
http://www.wolframalpha.com/input/?i=calculate+24%C3%B7(3%C3%B74)

As to "solving the equation", since you did not have an equation, there is nothing to solve