To find the difference between the first two terms:
[tex]\[
\frac{x}{x^2 - 16} - \frac{3}{x - 4},
\][/tex]
we'll proceed step-by-step.
### Step 1: Factorize the denominator
Firstly, notice that [tex]\(x^2 - 16\)[/tex] can be factored as:
[tex]\[
x^2 - 16 = (x + 4)(x - 4).
\][/tex]
Thus the first term becomes:
[tex]\[
\frac{x}{x^2 - 16} = \frac{x}{(x + 4)(x - 4)}.
\][/tex]
### Step 2: Rewrite the second fraction
The second term is:
[tex]\[
\frac{3}{x - 4}.
\][/tex]
To combine this with the first term, we'll write it with a common denominator. Since the common denominator is [tex]\( (x + 4)(x - 4) \)[/tex], we rewrite the second term as:
[tex]\[
\frac{3}{x - 4} = \frac{3(x + 4)}{(x + 4)(x - 4)}.
\][/tex]
### Step 3: Subtract the fractions
Now we have the two fractions with a common denominator:
[tex]\[
\frac{x}{(x + 4)(x - 4)} - \frac{3(x + 4)}{(x + 4)(x - 4)}.
\][/tex]
Combine them into a single fraction:
[tex]\[
\frac{x - 3(x + 4)}{(x + 4)(x - 4)}.
\][/tex]
### Step 4: Simplify the numerator
Simplify the numerator by distributing and combining like terms:
[tex]\[
x - 3(x + 4) = x - 3x - 12 = -2x - 12.
\][/tex]
### Step 5: Write final expression
The simplified form of the difference is:
[tex]\[
\frac{-2x - 12}{(x + 4)(x - 4)}.
\][/tex]
We can also factor out a common factor of [tex]\( -2 \)[/tex] from the numerator:
[tex]\[
\frac{-2(x + 6)}{(x + 4)(x - 4)}.
\][/tex]
Thus, the difference,
[tex]\[
\frac{x}{x^2-16} - \frac{3}{x-4}
\][/tex]
is:
[tex]\[
\frac{2(-x-6)}{x^2 - 16}.
\][/tex]
Hence, the result is:
[tex]\[
\boxed{\frac{2(-x - 6)}{x^2-16}}.
\][/tex]
