Solve `9x^(2) + 12 x + 4 le 0`

Answer

To solve the inequality \( 9x^2 + 12x + 4 \leq 0 \), we can follow these steps:

### Step 1: Factor the quadratic expression
We start with the quadratic expression \( 9x^2 + 12x + 4 \). We can factor this expression.

Notice that:
\[ 9x^2 + 12x + 4 = (3x + 2)^2 \]

### Step 2: Set the factored expression less than or equal to zero
Now we rewrite the inequality using the factored form:
\[ (3x + 2)^2 \leq 0 \]

### Step 3: Analyze the squared term
The expression \( (3x + 2)^2 \) is a square of a real number. The square of any real number is always non-negative (i.e., it is either zero or positive). Therefore, the only time this expression can be less than or equal to zero is when it equals zero:
\[ (3x + 2)^2 = 0 \]

### Step 4: Solve for \( x \)
To find the value of \( x \) that makes the expression equal to zero, we solve:
\[ 3x + 2 = 0 \]
Subtracting 2 from both sides gives:
\[ 3x = -2 \]
Dividing by 3 yields:
\[ x = -\frac{2}{3} \]

### Step 5: Write the solution set
Since the inequality is less than or equal to zero, the solution set includes the point where the expression equals zero. Therefore, the solution to the inequality is:
\[ x = -\frac{2}{3} \]

### Final Answer
The solution set is:
\[ x \in \left[-\frac{2}{3}, -\frac{2}{3}\right] \]
or simply:
\[ x = -\frac{2}{3} \]

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