Let's solve the given expression step by step.

Given:
$f(x) = 2x + 3$

We need to find the value of:

$[f(x)]^2 - 3f(x + 1)$

Step 1: Calculate $[f(x)]^2$

$f(x) = 2x + 3$ $[f(x)]^2 = (2x + 3)^2$ Expanding $(2x + 3)^2$: $[f(x)]^2 = (2x + 3)(2x + 3) = 4x^2 + 12x + 9$

Step 2: Calculate $f(x + 1)$

Substitute $x + 1$ into the function $f(x)$: $f(x + 1) = 2(x + 1) + 3 = 2x + 2 + 3 = 2x + 5$

Step 3: Calculate $-3f(x + 1)$

$-3f(x + 1) = -3(2x + 5) = -6x - 15$

Step 4: Combine the results

Now, combine the expressions from Step 1 and Step 3: $[f(x)]^2 - 3f(x + 1) = (4x^2 + 12x + 9) + (-6x - 15)$ $= 4x^2 + 12x - 6x + 9 - 15$ $= 4x^2 + 6x - 6$

Final Answer

The correct answer is:

(A) $4x^2 + 6x - 6$

Would you like further details or have any questions?

Here are five related questions to explore:

1. How do you find the derivative of the function $f(x) = 2x + 3$?
2. What is the general form of a quadratic equation?
3. How do you expand the square of a binomial, such as $(ax + b)^2$?
4. How would the expression change if $f(x) = 3x + 4$?
5. What are the steps to simplify an expression with multiple terms?

Tip: When expanding binomials, remember to apply the distributive property carefully to ensure all terms are accounted for.