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# multiply and simplify

multiply and simplify (x^2+x-12/x-3)*(x^2-2x-3/x^2+5x+4)

((x2 + x - 12)/(x - 3))·((x2 - 2x - 3)/(x2 + 5x + 4))

First, if possible, factor the polynomial(s) in each rational expression:

(x2 + x - 12)/(x - 3) = ((x + 4)(x - 3))/(x - 3)

Notice that in this rational expression there is a common factor in both the numerator and the denominator, that being  'x - 3'. With that, these 2 factors cancel each other out.

((x + 4)(x - 3))/(x - 3) = (x + 4)

For the second rational expression,

(x2 - 2x - 3)/(x2 + 5x + 4) = ((x + 1)(x - 3))/((x + 1)(x + 4))

Common factor in both numerator and denominator here is 'x + 1'

((x + 1)(x - 3))/((x + 1)(x + 4)) = (x - 3)/(x + 4)

Substitute the simplified forms of the rational expressions into the original problem:

((x2 + x - 12)/(x - 3))·((x2 - 2x - 3)/(x2 + 5x + 4))

= (x + 4)·((x - 3)/(x + 4))

= ((x + 4)(x - 3))/(x + 4)

After multiplying the simplified forms of the rational expressions, we see that there is another common factor in both the numerator and the denominator that cancels out, that being 'x + 4'. And that leaves us with the final solution:

((x + 4)(x - 3))/(x + 4) = (x - 3)

I'm going to assume the following problem is what you meant.  Let me know if not.

(x2+x-12)/(x-3) * (x2-2x-3)/(x2+5x+4)

This is a question really testing your ability to factor the polynomials involved.  Once you do that it is fairly easy.  Here's what it looks like after factoring each of the polynomials:

[(x-3)(x+4)]/(x-3) * [(x-3)(x+1)]/[(x+4)(x+1)]

If you write this out on paper it is easier to see what to do.  In the first fraction you'll see that there's an (x-3) in the top and the bottom, so they cancel to 1, leaving only an (x+4) in the top.  The second fraction has an (x+1) in the top and bottom which also cancels to 1.  This leaves us with (x+4)(x-3) in the top of the new fraction and (x+4) in the bottom.  The two (x+4) factors cancel to 1 leaving you with the very simple final answer of (x-3).

Write it out on paper and it will be easier to follow what I've said here.  Hope that helps.

Charlie Gibson