To solve the equation
\[ \frac{3}{16x + 32} = x + 2, \]
start by cross-multiplying to eliminate the fraction:
\[ 3 = (x + 2)(16x + 32). \]
Now expand the right side:
\[ 3 = 16x^2 + 32x + 32x + 64. \]
This simplifies to:
\[ 3 = 16x^2 + 64x + 64. \]
Next, rearranging the equation gives:
\[ 16x^2 + 64x + 64 - 3 = 0, \]
which simplifies to:
\[ 16x^2 + 64x + 61 = 0. \]
Now we will apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
In our case, \( a = 16 \), \( b = 64 \), and \( c = 61 \).
First, calculate the discriminant:
\[ b^2 - 4ac = 64^2 - 4 \cdot 16 \cdot 61. \]
Calculating step by step:
\[ 64^2 = 4096 \]
\[ 4 \cdot 16 \cdot 61 = 976, \]
so:
\[ b^2 - 4ac = 4096 - 976 = 3120. \]
Now, the roots are:
\[ x = \frac{-64 \pm \sqrt{3120}}{32}. \]
Calculating \( \sqrt{3120} \):
\[ \sqrt{3120} \approx 55.8. \]
Thus, substituting this back into the formula:
\[ x = \frac{-64 \pm 55.8}{32}. \]
Calculating both potential roots:
\[ x_1 = \frac{-64 + 55.8}{32} = \frac{-8.2}{32} \approx -0.25625, \]
\[ x_2 = \frac{-64 - 55.8}{32} = \frac{-119.8}{32} \approx -3.74375. \]
After calculating, we see no integers from initial options match our approximated solutions, suggesting a reevaluation of our steps might be needed.
To summarize, let's analyze which was requested:
- Simplified, we tackled the algebra, finding quadratic solutions leading to non-integer values, eliminating options leading back to the conclusion that further exploration was required possibly on assumptions.
- Matchups of \( (−4,−6),(−2,2)\) lead nowhere approachable compared to the nature of the function dictated broadly by our non-inset approximations.
None of the provided answers \( A, B, C, D \) appear directly relatable to the workings shown above under a full evaluation in rational terms.
Your final answer remains under filtered checks. Please provide clarification or confirm underlying concepts for further refinement.
1 answer