To find and simplify [tex]\(\left(\frac{f}{g}\right)(-2)\)[/tex] given the functions [tex]\(f(x) = x(-2x - 3)\)[/tex] and [tex]\(g(x) = -2x - 3\)[/tex], let's go through the steps one by one:
1. Evaluate [tex]\(f(-2)\)[/tex]:
- Substitute [tex]\(x = -2\)[/tex] into the function [tex]\(f(x) = x(-2x - 3)\)[/tex].
- We get [tex]\(f(-2) = -2(-2(-2) - 3)\)[/tex].
- Simplify the expression inside the parentheses: [tex]\(-2(-2) = 4\)[/tex], so [tex]\(-2(4) - 3 = -8 - 3 = -11\)[/tex].
- Then [tex]\(f(-2) = -2 \cdot (-11) = 22\)[/tex].
2. Evaluate [tex]\(g(-2)\)[/tex]:
- Substitute [tex]\(x = -2\)[/tex] into the function [tex]\(g(x) = -2x - 3\)[/tex].
- We get [tex]\(g(-2) = -2(-2) - 3\)[/tex].
- Simplify: [tex]\(-2 \cdot (-2) = 4\)[/tex], so [tex]\(4 - 3 = 1\)[/tex].
- Therefore, [tex]\(g(-2) = 1\)[/tex].
3. Compute [tex]\(\left(\frac{f}{g}\right)(-2)\)[/tex]:
- With [tex]\(f(-2) = 22\)[/tex] and [tex]\(g(-2) = 1\)[/tex],
- We have [tex]\(\left(\frac{f}{g}\right)(-2) = \frac{f(-2)}{g(-2)} = \frac{22}{1} = 22\)[/tex].
So the simplified result of [tex]\(\left(\frac{f}{g}\right)(-2)\)[/tex] is:
[tex]\[ \boxed{22} \][/tex]
