Question

what is the product?
\\(\left(x^{4}\
ight)\left(3x^{3}-2\
ight)\left(4x^{2}+5x\
ight)\\)?
\\(\bigcirc\\) \\(12x^{9}+15x^{8}-8x^{6}-10x^{5}\\)
\\(\bigcirc\\) \\(12x^{24}+15x^{12}-8x^{8}-10x^{4}\\)
\\(\bigcirc\\) \\(12x^{9}-10x^{5}\\)
\\(\bigcirc\\) \\(12x^{24}-10x^{4}\\)

Explanation:

Step1: Multiply the first two factors

First, multiply \(x^4\) with \((3x^3 - 2)\). Using the distributive property \(a(b + c)=ab+ac\), we have:
\(x^4(3x^3 - 2)=3x^{4 + 3}-2x^4 = 3x^7-2x^4\)

Step2: Multiply the result with the third factor

Now, multiply \((3x^7 - 2x^4)\) with \((4x^2 + 5x)\). Again, using the distributive property (FOIL method for binomials, extended for polynomials):
\[

$$\begin{align*} &(3x^7)(4x^2)+(3x^7)(5x)+(-2x^4)(4x^2)+(-2x^4)(5x)\\ =&12x^{7 + 2}+15x^{7+1}-8x^{4 + 2}-10x^{4+1}\\ =&12x^9 + 15x^8-8x^6-10x^5 \end{align*}$$

\]

Answer:

\(12x^9 + 15x^8-8x^6-10x^5\) (which corresponds to the first option: \(12x^9 + 15x^8 - 8x^6 - 10x^5\))