Simplify completely quantity 6 x squared minus 54 plus 84 over quantity 8 x squared minus 40 x plus 48 divided by quantity x squared plus x minus 56 over quantity 2 x squared plus 12 x minus 32

To simplify this expression, we need to follow the order of operations, which is commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). Let's break down the steps to simplify this expression:

Step 1: Simplify each numerator and denominator individually.
- For the numerator:
- 6x^2 - 54 + 84 simplifies to 6x^2 + 30.
- Similarly, for the denominator of the numerator:
- 8x^2 - 40x + 48 simplifies to 8x^2 - 40x + 48.
- For the denominator:
- x^2 + x - 56 cannot be simplified further.
- Similarly, for the denominator of the denominator:
- 2x^2 + 12x - 32 simplifies to 2x^2 + 12x - 32.

Step 2: Rewrite the expression using the simplified forms of the numerator and denominator:
- The simplified expression becomes (6x^2 + 30) / (8x^2 - 40x + 48) divided by (x^2 + x - 56) / (2x^2 + 12x - 32).

Step 3: Flip the second fraction (the one after the "divided by" sign) and change the division operation to multiplication:
- The expression becomes (6x^2 + 30) / (8x^2 - 40x + 48) * (2x^2 + 12x - 32) / (x^2 + x - 56).

Step 4: Factor the quadratics if possible:
- We can factor some of the quadratic expressions to simplify further:
- The denominator (8x^2 - 40x + 48) factors to 8(x^2 - 5x + 6), which can be simplified as 8(x - 2)(x - 3).
- The numerator (2x^2 + 12x - 32) factors to 2(x^2 + 6x - 16), which cannot be simplified further.
- The denominator (x^2 + x - 56) factors to (x + 8)(x - 7), which cannot be simplified further.

Step 5: Cancel out any common factors between the numerator and denominator in each fraction:
- The factors (x - 2) and (x + 8) exist in both the numerator and denominator. Canceling them out leaves us with:
- (3x + 15) / (8(x - 3)) * (2(x^2 + 6x - 16)) / (x - 7).

Step 6: Multiply the remaining factors together:
- The expression becomes (3x + 15) * (2(x^2 + 6x - 16)) / (8(x - 3)(x - 7)).

Step 7: Simplify any remaining terms if possible:
- (3x + 15) simplifies to 3(x + 5).
- 2(x^2 + 6x - 16) cannot be simplified further.

Step 8: Rewrite the expression using the simplified forms of the numerator and denominator:
- The simplified expression is (3(x + 5))(2(x^2 + 6x - 16)) / (8(x - 3)(x - 7)).

This is the completely simplified form of the expression.

To simplify the expression, let's break it down step by step.

Step 1: Simplify the numerator of the fraction on the left side, 6x^2 - 54 + 84.
This can be simplified as follows:
6x^2 - 54 + 84 = 6x^2 + 30

Step 2: Simplify the denominator of the fraction on the left side, 8x^2 - 40x + 48.
This can be simplified as follows:
8x^2 - 40x + 48 = 8(x^2 - 5x + 6) = 8(x - 3)(x - 2)

Step 3: Simplify the numerator of the fraction on the right side, x^2 + x - 56.
The numerator cannot be simplified further.

Step 4: Simplify the denominator of the fraction on the right side, 2x^2 + 12x - 32.
This can be simplified as follows:
2x^2 + 12x - 32 = 2(x^2 + 6x - 16) = 2(x + 8)(x - 2)

Now, the original expression can be written as:
(6x^2 + 30) / (8(x - 3)(x - 2)) ÷ (x^2 + x - 56) / (2(x + 8)(x - 2))

Step 5: Invert and multiply. Divide a fraction by another fraction by multiplying the first fraction by the reciprocal of the second fraction.
(6x^2 + 30) / (8(x - 3)(x - 2)) * (2(x + 8)(x - 2)) / (x^2 + x - 56)

Step 6: Cancel out common factors in the numerator and denominator, if possible.
The expression can be simplified further if there are any common factors between the numerator and denominator. However, in this case, there are no common factors that can be canceled out.

Thus, the simplified form of the expression is:
(6x^2 + 30)(2(x + 8)(x - 2)) / (8(x - 3)(x - 2))(x^2 + x - 56)