Alright, let's solve the problem step-by-step.
Given:
[tex]\[ f(x) = 2x - 3 \][/tex]
[tex]\[ g(x) = x^2 + 1 \][/tex]
We need to find the product of these two functions:
[tex]\[ (fg)(x) = f(x) \cdot g(x) \][/tex]
Now, substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (fg)(x) = (2x - 3) \cdot (x^2 + 1) \][/tex]
Next, we will expand this product using the distributive property:
[tex]\[ (2x - 3) \cdot (x^2 + 1) \][/tex]
Distribute [tex]\( 2x \)[/tex]:
[tex]\[ 2x \cdot x^2 + 2x \cdot 1 = 2x^3 + 2x \][/tex]
Distribute [tex]\( -3 \)[/tex]:
[tex]\[ -3 \cdot x^2 + (-3) \cdot 1 = -3x^2 - 3 \][/tex]
Now, combine all the terms:
[tex]\[ 2x^3 + 2x - 3x^2 - 3 \][/tex]
Arrange the terms in descending order of the powers of [tex]\( x \)[/tex]:
[tex]\[ 2x^3 - 3x^2 + 2x - 3 \][/tex]
Thus, the simplified form of [tex]\( (fg)(x) \)[/tex] is:
[tex]\[ \boxed{2x^3 - 3x^2 + 2x - 3} \][/tex]
