f(x)=(4x^2) +2x−5 and g(x)=(x^2) −3x+4, find (f−g)(x)

To find \((f - g)(x)\), we need to subtract \(g(x)\) from \(f(x)\). Here’s the step-by-step solution:

Given:
\[
f(x) = 4x^2 + 2x - 5
\]
\[
g(x) = x^2 - 3x + 4
\]

Then,
\[
(f - g)(x) = f(x) - g(x)
\]
\[
= (4x^2 + 2x - 5) - (x^2 - 3x + 4)
\]

Now distribute the negative sign across \(g(x)\):
\[
= 4x^2 + 2x - 5 - x^2 + 3x - 4
\]

Combine like terms:
\[
= (4x^2 - x^2) + (2x + 3x) + (-5 - 4)
\]
\[
= 3x^2 + 5x - 9
\]

So,
\[
(f - g)(x) = 3x^2 + 5x - 9
\]

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Would you like more details or have any questions? 

Here are five related questions you might find useful:
1. What is \((f + g)(x)\) for the same functions?
2. How do we find \((f \cdot g)(x)\) using \(f(x)\) and \(g(x)\)?
3. How does this process change if you need \((f/g)(x)\)?
4. What happens to \((f - g)(x)\) if we change the constants in \(f(x)\) and \(g(x)\)?
5. How would the graph of \((f - g)(x)\) compare to the graphs of \(f(x)\) and \(g(x)\)?

**Tip:** Always remember to distribute negative signs carefully when subtracting polynomials.

Calculate (f - g)(x) for functions f(x) = 4x^2 + 2x - 5 and g(x) = x^2 - 3x + 4 with a step-by-step breakdown. This algebra problem involves polynomial subtraction, function operations, and combining like terms, ideal for Grades 9-10.

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