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 the numerators and denominators.
For the first fraction, factor the numerator:
[tex]$$
x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x - 2)(x + 2),
$$[/tex]
and factor the denominator:
[tex]$$
5x^5 + 45x^4 + 70x^3 = 5x^3\left(x^2 + 9x + 14\right).
$$[/tex]
Notice that the quadratic factors as
[tex]$$
x^2 + 9x + 14 = (x + 7)(x + 2).
$$[/tex]
Thus, the denominator becomes
[tex]$$
5x^3 (x + 7)(x + 2).
$$[/tex]
For the second fraction, factor the numerator:
[tex]$$
5x + 35 = 5(x + 7),
$$[/tex]
with the denominator remaining as [tex]$x + 10$[/tex].
Step 2. Rewrite the expression with these factorizations.
Substitute the factored forms 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 the common factor [tex]$(x + 2)$[/tex] in the numerator and denominator.
2. Cancel the common factor [tex]$(x + 7)$[/tex] in the numerator and denominator.
3. Cancel the constant [tex]$5$[/tex] in the numerator and denominator.
4. Cancel [tex]$x^2$[/tex] in the numerator with part of [tex]$x^3$[/tex] in the denominator, leaving [tex]$x$[/tex] in the denominator.
After cancellation, the expression simplifies to:
[tex]$$
\frac{x - 2}{x(x + 10)}.
$$[/tex]
Final Answer:
[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 the numerators and denominators.
For the first fraction, factor the numerator:
[tex]$$
x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x - 2)(x + 2),
$$[/tex]
and factor the denominator:
[tex]$$
5x^5 + 45x^4 + 70x^3 = 5x^3\left(x^2 + 9x + 14\right).
$$[/tex]
Notice that the quadratic factors as
[tex]$$
x^2 + 9x + 14 = (x + 7)(x + 2).
$$[/tex]
Thus, the denominator becomes
[tex]$$
5x^3 (x + 7)(x + 2).
$$[/tex]
For the second fraction, factor the numerator:
[tex]$$
5x + 35 = 5(x + 7),
$$[/tex]
with the denominator remaining as [tex]$x + 10$[/tex].
Step 2. Rewrite the expression with these factorizations.
Substitute the factored forms 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 the common factor [tex]$(x + 2)$[/tex] in the numerator and denominator.
2. Cancel the common factor [tex]$(x + 7)$[/tex] in the numerator and denominator.
3. Cancel the constant [tex]$5$[/tex] in the numerator and denominator.
4. Cancel [tex]$x^2$[/tex] in the numerator with part of [tex]$x^3$[/tex] in the denominator, leaving [tex]$x$[/tex] in the denominator.
After cancellation, the expression simplifies to:
[tex]$$
\frac{x - 2}{x(x + 10)}.
$$[/tex]
Final Answer:
[tex]$$
\frac{x - 2}{x(x + 10)}.
$$[/tex]
