# How do you simplify #(2x^2+6x+4)/(4x^2-12x-16)#?

To make progress, attempts should be made to factorise the expressions on the top of the fraction (the numerator) and the bottom of the fraction (the denominator) to see if there are any in common that might be cancelled.

I will leave cancelling of any common factors until the end.

Next, it is possible to factorise both of the quadratics in the numerator and denominator as follows:

So, now the fun starts cancelling common factors from the numerator and the denominator.

implies

which may be written

There are no more common factors to be cancelled.

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To simplify the expression (2x^2+6x+4)/(4x^2-12x-16), we can first factor both the numerator and denominator. The numerator can be factored as 2(x+2)(x+1), and the denominator can be factored as 4(x-2)(x+2).

Next, we can cancel out any common factors between the numerator and denominator. In this case, we can cancel out the (x+2) term.

After canceling out the common factor, the simplified expression becomes 2(x+1)/(4(x-2)).

This can be further simplified by dividing both the numerator and denominator by 2, resulting in (x+1)/(2(x-2)).

Therefore, the simplified form of (2x^2+6x+4)/(4x^2-12x-16) is (x+1)/(2(x-2)).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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