Answer :
We start with the expression
[tex]$$
\frac{x^4 - 4x^2}{5x^5 + 45x^4 + 70x^3} \cdot \frac{5x + 35}{x + 10}.
$$[/tex]
Step 1. Factor each part
1. Factor the numerator of the first fraction:
[tex]$$
x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x-2)(x+2).
$$[/tex]
2. Factor the denominator of the first fraction:
[tex]$$
5x^5 + 45x^4 + 70x^3 = 5x^3(x^2 + 9x + 14) = 5x^3(x+7)(x+2).
$$[/tex]
3. Factor the numerator of the second fraction:
[tex]$$
5x + 35 = 5(x+7).
$$[/tex]
The denominator [tex]$x+10$[/tex] is already in simplest form.
Step 2. Rewrite the Expression
Now, substitute the factors into the original expression:
[tex]$$
\frac{x^2(x-2)(x+2)}{5x^3(x+7)(x+2)} \cdot \frac{5(x+7)}{x+10}.
$$[/tex]
Step 3. Cancel Common Factors
1. Cancel [tex]$(x+2)$[/tex] from the numerator and denominator:
[tex]$$
\frac{x^2(x-2)\,\cancel{(x+2)}}{5x^3(x+7)\,\cancel{(x+2)}} \cdot \frac{5(x+7)}{x+10}.
$$[/tex]
2. Cancel [tex]$(x+7)$[/tex] from the remaining factors:
[tex]$$
\frac{x^2(x-2)}{5x^3} \cdot \frac{5}{x+10}.
$$[/tex]
3. Cancel the common factor [tex]$5$[/tex] in the numerator and denominator:
[tex]$$
\frac{x^2(x-2)}{x^3} \cdot \frac{1}{x+10}.
$$[/tex]
4. Simplify [tex]$\frac{x^2}{x^3}$[/tex] to get [tex]$\frac{1}{x}$[/tex] (assuming [tex]$x \neq 0$[/tex]):
[tex]$$
\frac{x-2}{x} \cdot \frac{1}{x+10} = \frac{x-2}{x(x+10)}.
$$[/tex]
Final Answer
The fully simplified expression is
[tex]$$
\frac{x-2}{x(x+10)}.
$$[/tex]
[tex]$$
\frac{x^4 - 4x^2}{5x^5 + 45x^4 + 70x^3} \cdot \frac{5x + 35}{x + 10}.
$$[/tex]
Step 1. Factor each part
1. Factor the numerator of the first fraction:
[tex]$$
x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x-2)(x+2).
$$[/tex]
2. Factor the denominator of the first fraction:
[tex]$$
5x^5 + 45x^4 + 70x^3 = 5x^3(x^2 + 9x + 14) = 5x^3(x+7)(x+2).
$$[/tex]
3. Factor the numerator of the second fraction:
[tex]$$
5x + 35 = 5(x+7).
$$[/tex]
The denominator [tex]$x+10$[/tex] is already in simplest form.
Step 2. Rewrite the Expression
Now, substitute the factors into the original expression:
[tex]$$
\frac{x^2(x-2)(x+2)}{5x^3(x+7)(x+2)} \cdot \frac{5(x+7)}{x+10}.
$$[/tex]
Step 3. Cancel Common Factors
1. Cancel [tex]$(x+2)$[/tex] from the numerator and denominator:
[tex]$$
\frac{x^2(x-2)\,\cancel{(x+2)}}{5x^3(x+7)\,\cancel{(x+2)}} \cdot \frac{5(x+7)}{x+10}.
$$[/tex]
2. Cancel [tex]$(x+7)$[/tex] from the remaining factors:
[tex]$$
\frac{x^2(x-2)}{5x^3} \cdot \frac{5}{x+10}.
$$[/tex]
3. Cancel the common factor [tex]$5$[/tex] in the numerator and denominator:
[tex]$$
\frac{x^2(x-2)}{x^3} \cdot \frac{1}{x+10}.
$$[/tex]
4. Simplify [tex]$\frac{x^2}{x^3}$[/tex] to get [tex]$\frac{1}{x}$[/tex] (assuming [tex]$x \neq 0$[/tex]):
[tex]$$
\frac{x-2}{x} \cdot \frac{1}{x+10} = \frac{x-2}{x(x+10)}.
$$[/tex]
Final Answer
The fully simplified expression is
[tex]$$
\frac{x-2}{x(x+10)}.
$$[/tex]
