To solve for the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] that make the given expression [tex]\(\frac{3x}{x+2} + \frac{4x}{x-2}\)[/tex] equivalent to [tex]\(\frac{ax^2 + bx}{x^2 + c}\)[/tex], we will perform the following steps:
1. Find a common denominator for the two fractions.
2. Combine the fractions and simplify the numerator.
3. Identify the coefficients that correspond to [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
### Step 1: Find a common denominator
The fractions are:
[tex]\[
\frac{3x}{x+2} \quad \text{and} \quad \frac{4x}{x-2}
\][/tex]
The common denominator will be [tex]\((x+2)(x-2)\)[/tex].
### Step 2: Combine the fractions and simplify
First, rewrite each fraction with the common denominator:
[tex]\[
\frac{3x}{x+2} = \frac{3x(x-2)}{(x+2)(x-2)} \quad \text{and} \quad \frac{4x}{x-2} = \frac{4x(x+2)}{(x+2)(x-2)}
\][/tex]
Now combine the fractions:
[tex]\[
\frac{3x(x-2) + 4x(x+2)}{(x+2)(x-2)}
\][/tex]
Expand the numerators:
[tex]\[
3x(x-2) = 3x^2 - 6x \quad \text{and} \quad 4x(x+2) = 4x^2 + 8x
\][/tex]
Add these together:
[tex]\[
3x^2 - 6x + 4x^2 + 8x = 7x^2 + 2x
\][/tex]
So, the combined fraction is:
[tex]\[
\frac{7x^2 + 2x}{x^2 - 4}
\][/tex]
### Step 3: Identify the coefficients
We need the expression [tex]\(\frac{ax^2 + bx}{x^2 + c}\)[/tex] to match the form of [tex]\(\frac{7x^2 + 2x}{x^2 -4}\)[/tex].
By comparing:
[tex]\[
ax^2 + bx \quad \text{with} \quad 7x^2 + 2x
\][/tex]
And:
[tex]\[
x^2 + c \quad \text{with} \quad x^2 - 4
\][/tex]
We find:
[tex]\[
a = 7, \quad b = 2, \quad \text{and} \quad c = -4
\][/tex]
### Final Answer
So, the values are:
[tex]\[
a = 7, \quad b = 2, \quad c = -4
\][/tex]
