To find the product of the three expressions [tex]\(\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)\)[/tex], we will follow a systematic approach of multiplying them step by step.
First, let's multiply the first two expressions:
1. [tex]\(\left(7x^2\right) \times \left(2x^3 + 5\right)\)[/tex]:
[tex]\[
= 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
= 14x^5 + 35x^2
\][/tex]
Next, we take the result of this multiplication and multiply it by the third expression:
2. [tex]\(\left(14x^5 + 35x^2\right) \times \left(x^2 - 4x - 9\right)\)[/tex]:
We apply the distributive property (also known as the FOIL method for polynomials) to multiply each term in the first polynomial by each term in the second polynomial.
[tex]\[
= \left(14x^5 + 35x^2\right) \left(x^2 - 4x - 9\right)
\][/tex]
Expanding this, we get:
[tex]\[
= 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
[tex]\[
+ 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
This simplifies to:
[tex]\[
= 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Hence, the product of the given expressions is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
