To solve the equation
[tex]\[
\frac{8}{2 x^2-9 x-5} - \frac{3}{2 x + 1} = \frac{2}{x - 5},
\][/tex]
we will approach the problem step by step.
Step 1: Simplify the terms on the left-hand side
First, we factorize the quadratic expression in the denominator [tex]\(2 x^2 - 9 x - 5\)[/tex]:
[tex]\[2 x^2 - 9 x - 5.\][/tex]
We look for two numbers that multiply to [tex]\( 2 \cdot (-5) = -10 \)[/tex] and add up to [tex]\(-9\)[/tex]. Those numbers are [tex]\(-10\)[/tex] and [tex]\(1\)[/tex]. We can use these numbers to split the middle term and factor by grouping:
[tex]\[2 x^2 - 9 x - 5 = 2 x^2 - 10 x + x - 5 = 2 x (x - 5) + 1 (x - 5) = (2 x + 1) (x - 5).\][/tex]
Thus,
[tex]\[
\frac{8}{2 x^2 - 9 x - 5} = \frac{8}{(2 x + 1) (x - 5)}.
\][/tex]
Step 2: Rewrite the original equation
So the original equation becomes:
[tex]\[
\frac{8}{(2 x + 1)(x - 5)} - \frac{3}{2 x + 1} = \frac{2}{x - 5}.
\][/tex]
Step 3: Common Denominator
Combine the fractions on the left-hand side over a common denominator:
The common denominator for [tex]\(\frac{8}{(2 x + 1)(x - 5)}\)[/tex] and [tex]\(\frac{3}{2 x + 1}\)[/tex] is [tex]\((2 x + 1)(x - 5)\)[/tex].
[tex]\[
\frac{8 - 3(x - 5)}{(2 x + 1)(x - 5)} = \frac{2}{x - 5}.
\][/tex]
Step 4: Simplify the numerator
Simplify the numerator on the left-hand side:
[tex]\[
8 - 3(x - 5) = 8 - 3x + 15 = 23 - 3x.
\][/tex]
So the equation becomes:
[tex]\[
\frac{23 - 3 x}{(2 x + 1)(x - 5)} = \frac{2}{x - 5}.
\][/tex]
Step 5: Equate the numerators
Since the denominators on both sides of the equation are [tex]\((x - 5)\)[/tex], we can cancel out [tex]\((x - 5)\)[/tex] from both sides, leaving us with:
[tex]\[
\frac{23 - 3 x}{2 x + 1} = 2.
\][/tex]
Step 6: Solve for [tex]\(x\)[/tex]
To solve this, we multiply both sides by [tex]\(2 x + 1\)[/tex]:
[tex]\[
23 - 3 x = 2(2 x + 1).
\][/tex]
Expand the right side:
[tex]\[
23 - 3 x = 4 x + 2.
\][/tex]
Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
23 - 2 = 4 x + 3 x.
\][/tex]
Simplify:
[tex]\[
21 = 7 x.
\][/tex]
Divide by 7:
[tex]\[
x = 3.
\][/tex]
Thus, the solution to the equation is:
[tex]\[
\boxed{3}.
\][/tex]
