Question

what is the difference?
\\(\frac{x}{x^2 - 2x - 15} - \frac{4}{x^2 + 2x - 35}\\)
\\(\bigcirc\\) \\(\frac{x^2 + 3x + 12}{(x - 3)(x - 5)(x + 7)}\\)
\\(\bigcirc\\) \\(\frac{x(x + 3 - 12)}{(x + 3)(x - 5)(x + 7)}\\)
\\(\bigcirc\\) \\(\frac{x^2 + 3x + 12}{(x + 3)(x - 5)(x + 7)}\\)
\\(\bigcirc\\) \\(\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}\\)

Explanation:

Step1: Factor Denominators

Factor \(x^2 - 2x - 15\): Find two numbers with product \(-15\) and sum \(-2\), which are \(-5\) and \(3\). So \(x^2 - 2x - 15=(x - 5)(x + 3)\).
Factor \(x^2 + 2x - 35\): Find two numbers with product \(-35\) and sum \(2\), which are \(7\) and \(-5\). So \(x^2 + 2x - 35=(x + 7)(x - 5)\).

Step2: Find LCD

The denominators are \((x - 5)(x + 3)\) and \((x + 7)(x - 5)\). The least common denominator (LCD) is \((x + 3)(x - 5)(x + 7)\).

Step3: Rewrite Fractions

Rewrite \(\frac{x}{(x - 5)(x + 3)}\) with LCD: Multiply numerator and denominator by \((x + 7)\):
\(\frac{x(x + 7)}{(x + 3)(x - 5)(x + 7)}\).

Rewrite \(\frac{4}{(x + 7)(x - 5)}\) with LCD: Multiply numerator and denominator by \((x + 3)\):
\(\frac{4(x + 3)}{(x + 3)(x - 5)(x + 7)}\).

Step4: Subtract Fractions

Subtract the two fractions:
\[ \begin{align*} \frac{x(x + 7)}{(x + 3)(x - 5)(x + 7)}-\frac{4(x + 3)}{(x + 3)(x - 5)(x + 7)}&=\frac{x(x + 7)-4(x + 3)}{(x + 3)(x - 5)(x + 7)}\\ &=\frac{x^2 + 7x - 4x - 12}{(x + 3)(x - 5)(x + 7)}\\ &=\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)} \end{align*} \] Wait, no—wait, let's recheck the subtraction. Wait, original problem: \(\frac{x}{x^2 - 2x - 15}-\frac{4}{x^2 + 2x - 35}\). Wait, when we expand \(x(x + 7)=x^2 + 7x\), and \(4(x + 3)=4x + 12\). Then subtraction: \(x^2 + 7x - (4x + 12)=x^2 + 7x - 4x - 12=x^2 + 3x - 12\)? Wait, no, the third option has \(+12\), fourth has \(-12\). Wait, maybe I made a sign error. Wait, no: \(\frac{x}{(x - 5)(x + 3)}-\frac{4}{(x - 5)(x + 7)}\). Wait, LCD is \((x + 3)(x - 5)(x + 7)\). So first fraction: \(x(x + 7)\), second: \(4(x + 3)\). So numerator: \(x^2 + 7x - 4x - 12 = x^2 + 3x - 12\)? But the third option is \(x^2 + 3x + 12\), fourth is \(x^2 + 3x - 12\). Wait, let's re-express the original problem. Wait, maybe I factored wrong. Wait, \(x^2 - 2x - 15\): \(x^2 - 2x - 15=(x - 5)(x + 3)\) (since \(-5 + 3=-2\), \(-5*3=-15\)). Correct. \(x^2 + 2x - 35=(x + 7)(x - 5)\) (7 + (-5)=2, 7*(-5)=-35). Correct. So LCD is \((x + 3)(x - 5)(x + 7)\). Then first fraction: \(\frac{x}{(x - 5)(x + 3)}=\frac{x(x + 7)}{(x + 3)(x - 5)(x + 7)}\). Second fraction: \(\frac{4}{(x - 5)(x + 7)}=\frac{4(x + 3)}{(x + 3)(x - 5)(x + 7)}\). Now subtract: \(x(x + 7)-4(x + 3)=x^2 + 7x - 4x - 12=x^2 + 3x - 12\). Wait, but the third option is \(x^2 + 3x + 12\), fourth is \(x^2 + 3x - 12\). Wait, maybe the original problem was addition? No, the problem says "difference". Wait, maybe I misread the signs. Wait, let's check the options again. The third option: \(\frac{x^2 + 3x + 12}{(x + 3)(x - 5)(x + 7)}\), fourth: \(\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}\). Wait, maybe I made a mistake in the sign when subtracting. Wait, \(\frac{x}{A}-\frac{4}{B}=\frac{x*B - 4*A}{A*B}\). So \(A=(x - 5)(x + 3)\), \(B=(x - 5)(x + 7)\). So numerator: \(x*(x + 7)-4*(x + 3)=x^2 + 7x - 4x - 12=x^2 + 3x - 12\). So the fourth option? Wait, no, the fourth option's denominator is \((x + 3)(x - 5)(x + 7)\), numerator \(x^2 + 3x - 12\). Wait, but let's check the options again. Wait, the third option has \(+12\), fourth \(-12\). Wait, maybe I messed up the factoring. Wait, no—wait, original denominator \(x^2 - 2x - 15\): \(x^2 - 2x - 15=(x - 5)(x + 3)\) (correct, because \(-5 + 3=-2\), \(-5*3=-15\)). \(x^2 + 2x - 35=(x + 7)(x - 5)\) (correct, \(7 + (-5)=2\), \(7*(-5)=-35\)). So LCD is \((x + 3)(x - 5)(x + 7)\). Then the subtraction: \(x(x + 7)=x^2 + 7x\), \(4(x + 3)=4x + 12\). Then \(x^2 + 7x - (4x + 12)=x^2 + 3x - 12\). So the fourth option is \(\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}\), which matches. Wait, but let me check the options again. The options are:

1. \(\frac{x^2 + 3x + 12}{(x - 3)(x - 5)(x + 7)}\) (denominator wrong, \((x - 3)\) instead of \((x + 3)\))
2. \(\frac{x(x + 3 - 12)}{(x + 3)(x - 5)(x + 7)}\) (incorrect numerator)
3. \(\frac{x^2 + 3x + 12}{(x + 3)(x - 5)(x + 7)}\) (numerator sign wrong)
4. \(\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}\) (correct)

Wait, so the correct answer is the fourth option: \(\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}\), which is the last option (the fourth circle). Wait, but let me re-express the steps. Wait, maybe I made a mistake in the sign. Wait, no—subtracting \(\frac{4}{...}\) from \(\frac{x}{...}\), so it's \(x(x + 7)-4(x + 3)\). Let's compute that again: \(x^2 + 7x - 4x - 12 = x^2 + 3x - 12\). Yes. So the fourth option is correct.

Answer:

\(\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}\) (the fourth option: \(\boldsymbol{\frac{x^2 + 3x - 12}{(x + 3)(x - 5)(x + 7)}}\))