Answer :
We start with the functions
[tex]$$
f(x) = 5x^2 \quad \text{and} \quad g(x) = x^3 + 2x^2 - 5x.
$$[/tex]
To find the product [tex]\(f(x) \cdot g(x)\)[/tex], we multiply these two functions:
[tex]$$
f(x) \cdot g(x) = 5x^2 \cdot (x^3 + 2x^2 - 5x).
$$[/tex]
Using the distributive property, we multiply [tex]\(5x^2\)[/tex] by each term inside the parentheses:
1. Multiply [tex]\(5x^2\)[/tex] and [tex]\(x^3\)[/tex]:
[tex]$$
5x^2 \cdot x^3 = 5x^{2+3} = 5x^5.
$$[/tex]
2. Multiply [tex]\(5x^2\)[/tex] and [tex]\(2x^2\)[/tex]:
[tex]$$
5x^2 \cdot 2x^2 = 10x^{2+2} = 10x^4.
$$[/tex]
3. Multiply [tex]\(5x^2\)[/tex] and [tex]\(-5x\)[/tex]:
[tex]$$
5x^2 \cdot (-5x) = -25x^{2+1} = -25x^3.
$$[/tex]
Now, combine the products:
[tex]$$
5x^5 + 10x^4 - 25x^3.
$$[/tex]
Thus, the product [tex]\(f(x) \cdot g(x)\)[/tex] is
[tex]$$
\boxed{5x^5 + 10x^4 - 25x^3}.
$$[/tex]
[tex]$$
f(x) = 5x^2 \quad \text{and} \quad g(x) = x^3 + 2x^2 - 5x.
$$[/tex]
To find the product [tex]\(f(x) \cdot g(x)\)[/tex], we multiply these two functions:
[tex]$$
f(x) \cdot g(x) = 5x^2 \cdot (x^3 + 2x^2 - 5x).
$$[/tex]
Using the distributive property, we multiply [tex]\(5x^2\)[/tex] by each term inside the parentheses:
1. Multiply [tex]\(5x^2\)[/tex] and [tex]\(x^3\)[/tex]:
[tex]$$
5x^2 \cdot x^3 = 5x^{2+3} = 5x^5.
$$[/tex]
2. Multiply [tex]\(5x^2\)[/tex] and [tex]\(2x^2\)[/tex]:
[tex]$$
5x^2 \cdot 2x^2 = 10x^{2+2} = 10x^4.
$$[/tex]
3. Multiply [tex]\(5x^2\)[/tex] and [tex]\(-5x\)[/tex]:
[tex]$$
5x^2 \cdot (-5x) = -25x^{2+1} = -25x^3.
$$[/tex]
Now, combine the products:
[tex]$$
5x^5 + 10x^4 - 25x^3.
$$[/tex]
Thus, the product [tex]\(f(x) \cdot g(x)\)[/tex] is
[tex]$$
\boxed{5x^5 + 10x^4 - 25x^3}.
$$[/tex]
