Certainly! Let's find \((f - g)(x)\) step-by-step.
Given the two functions:
[tex]\[
f(x) = 2x^2 + 4x - 5
\][/tex]
[tex]\[
g(x) = 6x^3 - 2x^2 + 3
\][/tex]
We need to find \((f - g)(x)\), which means we subtract \(g(x)\) from \(f(x)\).
Let's start by writing out the expression for \((f - g)(x)\):
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
Substitute the given functions into this expression:
[tex]\[
(f - g)(x) = (2x^2 + 4x - 5) - (6x^3 - 2x^2 + 3)
\][/tex]
Now distribute the negative sign to each term in \(g(x)\):
[tex]\[
(f - g)(x) = 2x^2 + 4x - 5 - 6x^3 + 2x^2 - 3
\][/tex]
Next, combine like terms. Start with the highest degree term and proceed to the lowest:
1. The highest degree term is \( -6x^3 \), and it appears only in \(g(x)\):
[tex]\[
-6x^3
\][/tex]
2. For the \(x^2\) terms, we have \(2x^2\) from both \(f(x)\) and the subtracted \(g(x)\):
[tex]\[
2x^2 + 2x^2 = 4x^2
\][/tex]
3. For the \(x\) terms, we have:
[tex]\[
4x
\][/tex]
4. Finally, the constant terms are \(-5\) from \(f(x)\) and \(-3\) from the subtracted \(g(x)\):
[tex]\[
-5 - 3 = -8
\][/tex]
Putting these all together, we get:
[tex]\[
(f - g)(x) = -6x^3 + 4x^2 + 4x - 8
\][/tex]
So, the result of \((f - g)(x)\) is:
[tex]\[
\boxed{-6x^3 + 4x^2 + 4x - 8}
\][/tex]
