ShmoopTube

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Try this shmoopy question on for size...

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Solve: The quantity a squared minus a minus 12 all over the quantity a squared plus 3a

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plus 2 divided by the quantity a plus 3 over the quantity a plus 2.

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And here are the potential answers...

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Well, there are really 2 major steps going on here.

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One: multiplying by the inverse -- so we have to flip the 2nd fraction around and make

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it times rather than divided by...

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And then two: We clearly have a factorable expression here.

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For the first one we have an a in each of these parts of the parentheses...

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...then we have to figure out what 2 numbers multiply together to get negative 12 and also

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add together to get negative 1.

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What are the factors of 12? 1 and 12? Yeah... but that won't work.

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2 and 6 don't work either, because when added, no matter which is positive and which

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is negative, they won't total "1."

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3 and 4 however... DO work.

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So we want the ADDED part to be a negative 1... negative 4 plus 3 is negative 1, and

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negative 4 times 3 is negative 12, so... we're good.

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The first part of our answer is quantity a plus 3 times quantity a minus 4.

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Now to the denominator.

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We need 2 numbers that when multiplied are 2... and when added are 3.

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Well... 2 only has 2 and 1 as factors... and luckily, if we add 2 and 1 we get 3. So the

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bottom is a quick calculation.

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It's quantity a plus 2 times quantity a plus 1.

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Now we just rewrite everything so it's fully factored and... cross-out-able.

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The quantity a plus 3 over the quantity a plus 3 becomes 1; we can trash it.

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Same deal with the quantity a plus 2.

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And we are left with just quantity a minus 4 over a plus 1...

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...and the answer is C.

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Phew.

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