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------------------------------------------------ Given [tex]f(x)=5x^2[/tex] and [tex]g(x)=x^3+2x^2-5x[/tex], what is [tex]f(x) \cdot g(x)[/tex]?

A. [tex]x^3+7x^2-5[/tex]
B. [tex]5x^6+10x^4-25x^2[/tex]
C. [tex]5x^5+10x^4-25x^3[/tex]
D. [tex]-x^3+3x^2+5x[/tex]

Sure, let's find the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] step-by-step.

We are given:
[tex]\[ f(x) = 5x^2 \][/tex]
[tex]\[ g(x) = x^3 + 2x^2 - 5x \][/tex]

To find [tex]\( f(x) \cdot g(x) \)[/tex], we multiply these two functions together:

[tex]\[ f(x) \cdot g(x) = (5x^2) \cdot (x^3 + 2x^2 - 5x) \][/tex]

Let's distribute [tex]\( 5x^2 \)[/tex] across each term inside the parentheses:

1. [tex]\( 5x^2 \cdot x^3 = 5x^{2+3} = 5x^5 \)[/tex]
2. [tex]\( 5x^2 \cdot 2x^2 = 5 \cdot 2 \cdot x^{2+2} = 10x^4 \)[/tex]
3. [tex]\( 5x^2 \cdot (-5x) = 5 \cdot (-5) \cdot x^{2+1} = -25x^3 \)[/tex]

Now, we combine these results:

[tex]\[ f(x) \cdot g(x) = 5x^5 + 10x^4 - 25x^3 \][/tex]

So, the product [tex]\( f(x) \cdot g(x) \)[/tex] is:

[tex]\[ \boxed{5x^5 + 10x^4 - 25x^3} \][/tex]

Now let's compare this with the options provided:

- [tex]\( x^3 + 7x^2 - 5 \)[/tex]
- [tex]\( 5x^6 + 10x^4 - 25x^2 \)[/tex]
- [tex]\( 5x^5 + 10x^4 - 25x^3 \)[/tex]
- [tex]\( -x^3 + 3x^2 + 5x \)[/tex]

The correct option is:

[tex]\[ \boxed{5x^5 + 10x^4 - 25x^3} \][/tex]