Question

Express in simplest form:  3x+18/4x+8. ÷. x^2−3/6x+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 \((\frac{3x+18}{4x+8}) \div (\frac{x^2-3}{6x+2})\), we start by rewriting it as a multiplication by the reciprocal:

\[
\frac{3x+18}{4x+8} \cdot \frac{6x+2}{x^2-3}
\]

Next, we will simplify each part.

1. **Simplifying \(3x + 18\)**:
\[
3x + 18 = 3(x + 6)
\]

2. **Simplifying \(4x + 8\)**:
\[
4x + 8 = 4(x + 2)
\]

3. **Simplifying \(6x + 2\)**:
\[
6x + 2 = 2(3x + 1)
\]

4. **Factoring \(x^2 - 3\)** (this is already in simplest form, it is a difference of squares):
\[
x^2 - 3 \text{ stays as it is.}
\]

Now we can rewrite the entire expression with these factorizations:

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

Next, we can combine and multiply these fractions:

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

Now the numerator becomes:

\[
6(x+6)(3x+1)
\]

And the denominator is:

\[
4(x+2)(x^2-3)
\]

At this stage, check for any possible cancellations.

1. No factors are common between the numerator and denominator that could be canceled at this point.

Therefore, we leave it as:

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

Finally, we can simplify \(6/4\) to \(3/2\):

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

This is the simplest form of the expression.

Thus, based on your options provided, none perfectly matches, but this is the simplest form of the expression.