given that f(x) = 3x + 2 and g(x) = x^2 - 1, find a fully simplified expression for g(x) = (x-1)(3x+1)-(x-1)^2

To simplify the given expression:

\[
g(x) = (x-1)(3x+1) - (x-1)^2
\]

let's break it down step by step.

### Step 1: Expand \((x-1)(3x+1)\)
Using the distributive property (FOIL method):
\[
(x-1)(3x+1) = x(3x+1) - 1(3x+1) = 3x^2 + x - 3x - 1
\]
Simplify:
\[
3x^2 + x - 3x - 1 = 3x^2 - 2x - 1
\]

### Step 2: Expand \((x-1)^2\)
\[
(x-1)^2 = (x-1)(x-1) = x^2 - 2x + 1
\]

### Step 3: Subtract the two expressions
Now subtract \((x-1)^2\) from \((x-1)(3x+1)\):
\[
g(x) = (3x^2 - 2x - 1) - (x^2 - 2x + 1)
\]
Distribute the negative sign:
\[
g(x) = 3x^2 - 2x - 1 - x^2 + 2x - 1
\]

### Step 4: Combine like terms
\[
g(x) = (3x^2 - x^2) + (-2x + 2x) + (-1 - 1)
\]
\[
g(x) = 2x^2 - 2
\]

Thus, the fully simplified expression for \(g(x)\) is:
\[
g(x) = 2x^2 - 2
\]

Would you like more details or have any questions?

### Related Questions:
1. How would you compute the derivative of \( g(x) = 2x^2 - 2 \)?
2. What is the graph of \( g(x) = 2x^2 - 2 \)?
3. How does the expansion of binomials relate to the solution?
4. Can you find the roots of \( g(x) = 2x^2 - 2 \)?
5. What is the value of \( g(x) \) when \( x = 1 \)?

**Tip**: When simplifying expressions, combining like terms carefully ensures accuracy!

Given that f(x) = 3x + 2 and g(x) = x^2 - 1, find a fully simplified expression for g(x) = (x-1)(3x+1) - (x-1)^2

This solution provides a detailed step-by-step process to simplify the algebraic expression g(x) = (x-1)(3x+1) - (x-1)^2. It explores key concepts such as polynomial expansion, binomial square, and simplification techniques. Suitable for students in Grades 9-12.

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