Riley makes a mistake in step 2 while doing her homework. What was her mistake \( \frac{x}{x^{2}-5 x+6}+\frac{3}{x+3} \)

Step 1: Factor the quadratic in the first denominator. We have \[ x^2-5x+6. \] This factors into \[ (x-2)(x-3). \] Step 2: Identify Riley’s mistake. Riley’s error in her homework came in step 2 when she incorrectly factored the quadratic \( x^2-5x+6 \) as \[ (x-2)(x+3) \] instead of the correct factorization \[ (x-2)(x-3). \] Because of this mistake, she would have incorrectly identified a common factor with the second denominator \( x+3 \) and proceeded to cancel it, which is not valid. Step 3: State the consequence. By erroneously assuming that \( x+3 \) is a factor of \( x^2-5x+6 \), she improperly simplified the expression. The correct approach is to recognize that no factor in the first denominator cancels with the \( x+3 \) from the second fraction. Thus, Riley’s mistake was misfactoring the quadratic and inappropriately canceling a term that does not appear in the actual factorization.