To find $g(f(x))$, we need to substitute $f(x) = x + 5$ into the function $g(x) = 3x^2 - 7x + 4$.

Step-by-step process:

1. Substitute $f(x) = x + 5$ into $g(x)$: $g(f(x)) = g(x + 5)$

2. Plug in $x + 5$ for $x$ in $g(x)$: $g(x + 5) = 3(x + 5)^2 - 7(x + 5) + 4$

3. Expand $(x + 5)^2$: $(x + 5)^2 = x^2 + 10x + 25$ So: $3(x + 5)^2 = 3(x^2 + 10x + 25) = 3x^2 + 30x + 75$

4. Distribute the -7: $-7(x + 5) = -7x - 35$

5. Now combine all terms: $g(f(x)) = 3x^2 + 30x + 75 - 7x - 35 + 4$

6. Simplify the expression: $g(f(x)) = 3x^2 + (30x - 7x) + (75 - 35 + 4)$ $g(f(x)) = 3x^2 + 23x + 44$

So, the polynomial expression for $g(f(x))$ in standard form is:

$g(f(x)) = 3x^2 + 23x + 44$

Would you like more details on any part of the process?

Relative Questions:

1. How do you expand binomials like $(x + 5)^2$?
2. Can you explain the role of substitution in function composition?
3. What is the general method for simplifying polynomials?
4. How does function composition work in different contexts, such as trigonometry?
5. How do you verify if a function composition is correct?

Tip: Always expand and simplify terms carefully when dealing with function compositions to avoid errors in coefficients.