Let's solve the equation [tex]\(\frac{x^2 - (x+1)(x+2)}{5x + 1} = 6\)[/tex] step by step.
Step 1: Simplify the numerator:
[tex]\[
x^2 - (x+1)(x+2)
\][/tex]
Expand [tex]\((x+1)(x+2)\)[/tex]:
[tex]\[
(x+1)(x+2) = x^2 + 3x + 2
\][/tex]
So, we have:
[tex]\[
x^2 - (x^2 + 3x + 2) = x^2 - x^2 - 3x - 2 = -3x - 2
\][/tex]
Step 2: Substitute the simplified numerator back into the equation:
[tex]\[
\frac{-3x - 2}{5x + 1} = 6
\][/tex]
Step 3: Clear the fraction by multiplying both sides of the equation by [tex]\(5x + 1\)[/tex]:
[tex]\[
-3x - 2 = 6(5x + 1)
\][/tex]
[tex]\[
-3x - 2 = 30x + 6
\][/tex]
Step 4: Move all terms involving [tex]\(x\)[/tex] to one side of the equation:
[tex]\[
-3x - 2 - 30x = 6
\][/tex]
[tex]\[
-3x - 30x - 2 = 6
\][/tex]
[tex]\[
-33x - 2 = 6
\][/tex]
Step 5: Isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-33x - 2 + 2 = 6 + 2
\][/tex]
[tex]\[
-33x = 8
\][/tex]
Step 6: Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{8}{-33}
\][/tex]
[tex]\[
x = -\frac{8}{33}
\][/tex]
Thus, the solution to the equation [tex]\(\frac{x^2 - (x+1)(x+2)}{5x + 1} = 6\)[/tex] is:
[tex]\[
x = -\frac{8}{33}
\][/tex]
