Question

what is the product?\\(\left(7x^{2}\
ight)\left(2x^{3}+5\
ight)\left(x^{2}-4x - 9\
ight)\\)\\(\bigcirc\\ 14x^{5}-x^{4}-46x^{3}-58x^{2}-20x - 45\\)\\(\bigcirc\\ 14x^{6}-56x^{5}-91x^{4}-140x^{3}-315x^{2}\\)\\(\bigcirc\\ 14x^{7}-56x^{6}-126x^{5}+35x^{4}-140x^{3}-315x^{2}\\)\\(\bigcirc\\ 14x^{12}-182x^{6}+35x^{4}-455x^{2}\\)

Explanation:

Step1: Multiply the first two factors

First, multiply \(7x^{2}\) and \(2x^{3}+5\). Using the distributive property (also known as the distributive law of multiplication over addition), we have:
\(7x^{2}(2x^{3}+5)=7x^{2}\times2x^{3}+7x^{2}\times5 = 14x^{5}+35x^{2}\)

Step2: Multiply the result by the third factor

Now, multiply \(14x^{5}+35x^{2}\) by \(x^{2}-4x - 9\). Again, using the distributive property (multiplying each term in the first polynomial by each term in the second polynomial):
\[

$$\begin{align*} &(14x^{5}+35x^{2})(x^{2}-4x - 9)\\ =&14x^{5}(x^{2}-4x - 9)+35x^{2}(x^{2}-4x - 9)\\ =&14x^{5}\times x^{2}-14x^{5}\times4x-14x^{5}\times9+35x^{2}\times x^{2}-35x^{2}\times4x-35x^{2}\times9\\ =&14x^{7}-56x^{6}-126x^{5}+35x^{4}-140x^{3}-315x^{2} \end{align*}$$

\]

Answer:

\(14x^{7}-56x^{6}-126x^{5}+35x^{4}-140x^{3}-315x^{2}\) (the third option)