Express in simplest form: 3x+18/4x+8÷x^2−36x+2 .(1 point)Responses4(x−6)3Start Fraction 4 left parenthesis x minus 6 right parenthesis over 3 End Fraction34x−8Start Fraction 3 over 4 x minus 8 End Fraction34x−24Start Fraction 2 over 4 x minus 24 End Fraction14(x−2)

Answer 1

To simplify the expression \( \frac{3x + 18}{4x + 8} \div (x^2 - 36x + 2) \), we will first rewrite the division as multiplication by the reciprocal:

\[ \frac{3x + 18}{4x + 8} \times \frac{1}{x^2 - 36x + 2}. \]

Next, let's simplify each component.

1. Factor the numerator \( 3x + 18 \): \[ 3x + 18 = 3(x + 6). \]

2. Factor the denominator \( 4x + 8 \): \[ 4x + 8 = 4(x + 2). \]

So we can rewrite the first fraction as:

\[ \frac{3(x + 6)}{4(x + 2)}. \]

3. Factor the quadratic expression \( x^2 - 36x + 2 \). This quadratic expression does not factor nicely, but we can find its roots using the quadratic formula: \[ x = \frac{-(-36) \pm \sqrt{(-36)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{36 \pm \sqrt{1296 - 8}}{2} = \frac{36 \pm \sqrt{1288}}{2}. \]

Since we want a simplified form instead of exact roots, we can leave it as is for now.

Putting it all back together, we have:

\[ \frac{3(x + 6)}{4(x + 2)} \times \frac{1}{x^2 - 36x + 2}. \]

Combining into a single fraction gives:

\[ \frac{3(x + 6)}{4(x + 2)(x^2 - 36x + 2)}. \]

In most cases for simplification, we look for similar factors to cancel out; however, \( x^2 - 36x + 2 \) does not seem to factor anymore conveniently with the terms we have.

Thus, our simplest form of the original expression is:

\[ \frac{3(x + 6)}{4(x + 2)(x^2 - 36x + 2)}. \]

Among the choices provided, none match this expression directly, suggesting it might have been simplified or presented with different numerical factors or constants.

Let's go through the options:

1. \( \frac{4(x - 6)}{3} \)
2. \( \frac{3}{4x - 8} \)
3. \( \frac{3}{4x - 24} \)
4. \( \frac{14(x - 2)}{} \)

However, since we're likely looking for a straightforward answer, and under basic manipulation, we observe that none of these straightforwardly yield the same base.

To finalize the solution, if we simplify to a most recognizable form based on common calculations in algebra, the resultant provided answer simplifies to be \( \frac{3(x + 6)}{4(x + 2)(x^2 - 36x + 2)} \), but without exact factors that could yield the answers provided.

Hence the valid selection should closely circle around being \( \frac{4(x-6)}{3} \) subject to additional simplifications yielding from polynomial behavior when evaluating divisions vs linear summarizations, which might warrant deeper review of the initial problem setup.

However, if forced to select the closest approximation to a finalized form simplifying broadly under the quadratic outputs, we might signal towards \( 4(x-6)/3 \) dependent on overarching simplifications erroneously derived in earlier algebra reductions.