Answer
$-3f(x) \cdot g(x) = -6x^3 + 3x^2 + 33x - 30$
$\text{The domain is the set of all real numbers.}$
Work Step by Step
This exercise asks us to multiply $-3$ times $f(x)$ to $g(x)$. Let's write out the problem:
$-3f(x) \cdot g(x) = -3(2x + 5)(x^2 - 3x + 2)$
Distribute terms first, paying attention to the signs:
$-3f(x) \cdot g(x) = -3[(2x)(x^2) + (2x)(-3x) + (2x)(2) + (5)(x^2) + (5)(-3x) + (5)(2)]$
Multiply out the terms:
$-3f(x) \cdot g(x) = -3[(2x^3) + (-6x^2) + (4x) + (5x^2) + (-15x) + (10)]$
Multiply all terms by $-3$:
$-3f(x) \cdot g(x) = -6x^3 + 18x^2 - 12x - 15x^2 + 45x - 30$
Combine like terms:
$-3f(x) \cdot g(x) = -6x^3 + 3x^2 + 33x - 30$
When we find the domain, we want to find which values of $x$ will cause the function to become undefined; in other words, we want to find any restrictions for $x$. In this exercise, $x$ can be any real number since there are no restrictions, so the domain is all real numbers.
