# How do you simplify #(6x^2-54x+84)/(8x^2-40x+48) div (x^2+x-56)/(x^2+12x+32)#?

Start by writing down your starting expression

All these quadratics can be easily factored by using the sum/product technique to get

The expression will thus be equal to

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To simplify the expression (6x^2-54x+84)/(8x^2-40x+48) divided by (x^2+x-56)/(x^2+12x+32), we can first factorize the numerator and denominator of both fractions:

(6x^2-54x+84) factors to 6(x-2)(x-7) (8x^2-40x+48) factors to 8(x-2)(x-3) (x^2+x-56) factors to (x-7)(x+8) (x^2+12x+32) factors to (x+4)(x+8)

Next, we can rewrite the expression as a multiplication by flipping the second fraction and multiplying:

(6(x-2)(x-7))/(8(x-2)(x-3)) * ((x+8)(x-7))/(x+4)(x+8)

Now, we can cancel out common factors:

(6)/(8(x-3)) * 1/(x+4)

Simplifying further:

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

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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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