We are given two functions:

- \( f(x) = 11x^2 - 3x \)
- \( g(x) = 10x + 4 \)

The expression we need to evaluate is:

\[
g(x)^2 - 3f(x)
\]

### Step 1: Calculate \( g(x)^2 \)

\[
g(x)^2 = (10x + 4)^2
\]

Expand the square:

\[
g(x)^2 = (10x)^2 + 2(10x)(4) + (4)^2 = 100x^2 + 80x + 16
\]

### Step 2: Calculate \( 3f(x) \)

\[
3f(x) = 3(11x^2 - 3x) = 33x^2 - 9x
\]

### Step 3: Subtract \( 3f(x) \) from \( g(x)^2 \)

\[
g(x)^2 - 3f(x) = (100x^2 + 80x + 16) - (33x^2 - 9x)
\]

Now simplify:

\[
g(x)^2 - 3f(x) = 100x^2 - 33x^2 + 80x + 9x + 16
\]

\[
g(x)^2 - 3f(x) = 67x^2 + 89x + 16
\]

So, the final expression is:

\[
g(x)^2 - 3f(x) = 67x^2 + 89x + 16
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the solution change if the coefficients in \( f(x) \) or \( g(x) \) were different?
2. What happens when \( g(x) \) is a quadratic function instead of linear?
3. How do you expand higher-degree polynomials like \( g(x)^3 \)?
4. Can we factor the final expression \( 67x^2 + 89x + 16 \)?
5. What is the significance of finding the roots of \( g(x)^2 - 3f(x) \)?

### Tip:
When expanding polynomials, keep track of terms by aligning like powers of \( x \) to simplify correctly!

f(x)=11x^2-3x, g(x)=10x+4. Solve g(x)^2 - 3f(x)

Explore the solution for g(x)^2 - 3f(x) where f(x) = 11x^2 - 3x and g(x) = 10x + 4. Learn the step-by-step process of expanding and simplifying polynomials to find the final expression. This problem covers key algebraic concepts such as polynomial functions, quadratic expressions, and combining like terms. Ideal for students in Grades 9-11.

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