To simplify the given expression step-by-step, we need to factorize each part of the expression and then simplify. The expression is:
[tex]\[
\frac{6 x^5 + 30 x^4 - 84 x^3}{3 x^5 - 30 x^4 + 48 x^3} \cdot \frac{x - 8}{x^2 + 17 x + 70}
\][/tex]
### Step 1: Factorize the Numerator and Denominator of the First Fraction
#### Numerator: [tex]\( 6 x^5 + 30 x^4 - 84 x^3 \)[/tex]
Factor out the greatest common factor (GCF), which is [tex]\( 6 x^3 \)[/tex]:
[tex]\[
6 x^5 + 30 x^4 - 84 x^3 = 6 x^3 (x^2 + 5x - 14)
\][/tex]
#### Denominator: [tex]\( 3 x^5 - 30 x^4 + 48 x^3 \)[/tex]
Factor out the GCF, which is [tex]\( 3 x^3 \)[/tex]:
[tex]\[
3 x^5 - 30 x^4 + 48 x^3 = 3 x^3 (x^2 - 10x + 16)
\][/tex]
Now the first fraction becomes:
[tex]\[
\frac{6 x^3 (x^2 + 5x - 14)}{3 x^3 (x^2 - 10x + 16)}
\][/tex]
Simplify by canceling the common term [tex]\( 3 x^3 \)[/tex]:
[tex]\[
\frac{6 (x^2 + 5x - 14)}{3 (x^2 - 10x + 16)} = 2 \cdot \frac{x^2 + 5x - 14}{x^2 - 10x + 16}
\][/tex]
### Step 2: Factor the Quadratics in the Simplified Expression
#### Numerator: [tex]\( x^2 + 5x - 14 \)[/tex]
Solve for the roots of the quadratic equation [tex]\( x^2 + 5x - 14 = 0 \)[/tex]:
[tex]\[
x = \frac{-5 \pm \sqrt{25 + 56}}{2} = \frac{-5 \pm \sqrt{81}}{2} = \frac{-5 \pm 9}{2}
\][/tex]
The roots are [tex]\( x = 2 \)[/tex] and [tex]\( x = -7 \)[/tex], so we can factor as:
[tex]\[
x^2 + 5x - 14 = (x - 2)(x + 7)
\][/tex]
#### Denominator: [tex]\( x^2 - 10x + 16 \)[/tex]
Solve for the roots of the quadratic equation [tex]\( x^2 - 10x + 16 = 0 \)[/tex]:
[tex]\[
x = \frac{10 \pm \sqrt{100 - 64}}{2} = \frac{10 \pm \sqrt{36}}{2} = \frac{10 \pm 6}{2}
\][/tex]
The roots are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex], so we can factor as:
[tex]\[
x^2 - 10x + 16 = (x - 8)(x - 2)
\][/tex]
Now the simplified expression becomes:
[tex]\[
2 \cdot \frac{(x - 2)(x + 7)}{(x - 8)(x - 2)}
\][/tex]
### Step 3: Simplify the Fraction by Cancelling Common Factors
Cancel the common factor [tex]\( x - 2 \)[/tex]:
[tex]\[
2 \cdot \frac{(x + 7)}{(x - 8)}
\][/tex]
### Step 4: Combine with the Second Fraction
Multiply by the second fraction [tex]\(\frac{x - 8}{x^2 + 17x + 70}\)[/tex]:
[tex]\[
2 \cdot \frac{(x + 7)}{(x - 8)} \cdot \frac{(x - 8)}{x^2 + 17x + 70}
\][/tex]
Cancel the common factor [tex]\( x - 8 \)[/tex]:
[tex]\[
2 \cdot \frac{(x + 7)}{x^2 + 17x + 70}
\][/tex]
### Step 5: Factorize [tex]\( x^2 + 17x + 70 \)[/tex]
Factor the quadratic:
[tex]\[
x^2 + 17x + 70 = (x + 7)(x + 10)
\][/tex]
So the final simplified expression is:
[tex]\[
2 \cdot \frac{x + 7}{(x + 7)(x + 10)}
\][/tex]
Cancel the common factor [tex]\( x + 7 \)[/tex]:
[tex]\[
2 \cdot \frac{1}{x + 10} = \frac{2}{x + 10}
\][/tex]
Thus, the fully simplified expression is:
[tex]\[
\boxed{\frac{2}{x + 10}}
\][/tex]
