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------------------------------------------------ Fully simplify the expression below and write your answer as a single fraction.

\[
\left(\frac{x^4 - 4x^2}{5x^5 + 45x^4 + 70x^3}\right) \times \left(\frac{5x + 36}{x + 10}\right)
\]

To simplify the given expression, we first factorize each polynomial in the expression. Factoring the numerator [tex]x^4[/tex] - 4[tex]x^2[/tex], we get

x²([tex]x^2[/tex] - 4),

which further simplifies to [tex]x^2[/tex](x - 2)(x + 2).

Factoring the denominator 5x⁵ + 45[tex]x^4[/tex] + 70[tex]x^3[/tex], we find that it has a common factor of 5[tex]x^3[/tex].

So, after factoring out 5[tex]x^3[/tex], we get 5[tex]x^3[/tex]([tex]x^2[/tex] + 9x + 14), which factors to 5[tex]x^3[/tex](x + 2)(x + 7).

Next, we factorize the second fraction's numerator 5x + 36 and denominator x + 10.

These do not further factorize. Now, we simplify by canceling out common factors between the numerators and denominators. We can cancel out the (x + 2) terms from both the numerator and denominator, leaving us with ([tex]x^2[/tex](x - 2) * 1) / (5[tex]x^3[/tex](x + 7) * 1).

Further simplifying, we cancel out x terms from the numerator and denominator, which results in

(x(x - 2)) / (5x(x + 7)).

Finally, we cancel out another x term from both numerator and denominator, yielding the simplified expression ([tex]x^3[/tex] - 4x) / ([tex]x^4[/tex] + 9[tex]x^3[/tex] + 14[tex]x^2[/tex]).