Let's solve the inequality step by step:

\[
-(3x + 2)(5 - x) - 3 \leq 3(x + 2)^2
\]

### Step 1: Expand both sides.

Start by expanding the left-hand side:
\[
-(3x + 2)(5 - x) = -(15 - 3x - 10x + 2x^2) = -(15 - 13x + 2x^2)
\]
\[
= -15 + 13x - 2x^2
\]
So the left-hand side becomes:
\[
-15 + 13x - 2x^2 - 3 = -18 + 13x - 2x^2
\]

Now expand the right-hand side:
\[
3(x + 2)^2 = 3(x^2 + 4x + 4) = 3x^2 + 12x + 12
\]

### Step 2: Set up the inequality.

Now the inequality is:
\[
-18 + 13x - 2x^2 \leq 3x^2 + 12x + 12
\]

### Step 3: Move all terms to one side.

Add \( 2x^2 - 13x + 18 \) to both sides:
\[
0 \leq 5x^2 - x + 30
\]

Or equivalently:
\[
5x^2 - x + 30 \geq 0
\]

### Step 4: Analyze the quadratic inequality.

Now we analyze \( 5x^2 - x + 30 \geq 0 \).

First, find the discriminant to check if it has real roots:
\[
\Delta = (-1)^2 - 4(5)(30) = 1 - 600 = -599
\]

Since the discriminant is negative, the quadratic \( 5x^2 - x + 30 \) has no real roots, meaning the expression is always positive (since the coefficient of \(x^2\) is positive). Therefore, the inequality is always true.

### Final Answer:
The inequality \( -(3x + 2)(5 - x) - 3 \leq 3(x + 2)^2 \) is true for all real values of \(x\).

Would you like a detailed breakdown of any of these steps? 

Here are 5 related questions:
1. What is the significance of the discriminant in quadratic equations?
2. How do you solve inequalities involving quadratic expressions?
3. What happens if the discriminant is zero?
4. How do you factor a quadratic expression?
5. What is the general method to expand binomials?

**Tip:** When solving inequalities involving quadratics, always check the discriminant to understand the nature of the roots (if any).

Step-by-step solution for solving the inequality (3x+2)(5-x)-3 <= 3*(x+2)^2. This problem explores algebraic expansion, quadratic inequalities, and discriminant analysis to determine if the inequality holds for all real values of x. Ideal for students in Grades 10-12.

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