Let's solve the inequality:

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

### Step 1: Expand both sides

**Left side:**

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

Use distributive property:

\[
-(15 - 3x^2 - 10x + 2x) = -(-3x^2 - 8x + 15)
\]

This simplifies to:

\[
3x^2 + 8x - 15
\]

Now, subtract 3 from this:

\[
3x^2 + 8x - 15 - 3 = 3x^2 + 8x - 18
\]

**Right side:**

Expand \(3(x + 2)^2\):

\[
3(x^2 + 4x + 4) = 3x^2 + 12x + 12
\]

### Step 2: Combine everything

Now, write the inequality:

\[
3x^2 + 8x - 18 \leq 3x^2 + 12x + 12
\]

### Step 3: Simplify

Subtract \(3x^2\) from both sides:

\[
8x - 18 \leq 12x + 12
\]

Subtract \(12x\) from both sides:

\[
-4x - 18 \leq 12
\]

Add 18 to both sides:

\[
-4x \leq 30
\]

### Step 4: Solve for \(x\)

Divide both sides by \(-4\) (remember to flip the inequality sign when dividing by a negative number):

\[
x \geq -\frac{30}{4}
\]

\[
x \geq -7.5
\]

### Final solution:
\[
x \geq -7.5
\]

### Do you have any questions, or would you like more details?

Here are 5 related questions for further exploration:
1. How do you solve a system of inequalities?
2. What happens if you divide an inequality by zero?
3. Can inequalities be solved using graphing techniques?
4. What are the properties of quadratic inequalities?
5. How do you interpret inequalities in real-world problems?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the inequality -(3x + 2)(5 - x) - 3 <= 3(x + 2)^2 with a detailed step-by-step solution. This problem covers essential algebraic concepts, including expanding binomials, solving quadratic inequalities, and applying inequality rules. Suitable for students in Grades 9-11.

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