Question

x² - 11x + 24
3x - 9 · 6x + 42
x² + 4x - 96

Explanation:

Step1: Factor numerators and denominators

Factor \(x^2 - 11x + 24\): Find two numbers that multiply to 24 and add to -11. They are -3 and -8. So \(x^2 - 11x + 24=(x - 3)(x - 8)\).
Factor \(3x - 9\): Factor out 3, so \(3x - 9 = 3(x - 3)\).
Factor \(6x + 42\): Factor out 6, so \(6x + 42 = 6(x + 7)\).
Factor \(x^2 + 14x - 96\): Find two numbers that multiply to -96 and add to 14. They are 16 and -6. So \(x^2 + 14x - 96=(x + 16)(x - 6)\)? Wait, no: \(16\times(-6)=-96\), \(16+(-6)=10\). Wait, mistake. Let's recalculate: looking for \(a\) and \(b\) such that \(a + b = 14\), \(ab=-96\). \(16\times(-6)=-96\), \(16-6=10\). No. Wait, \(24\times(-4)=-96\), \(24-4=20\). No. Wait, \(12\times(-8)=-96\), \(12-8=4\). No. Wait, original problem: maybe a typo? Wait, the second denominator is \(x^2 + 14x - 96\)? Wait, maybe it's \(x^2 + 14x - 96\) or maybe \(x^2 + 14x - 96\) is correct? Wait, no, let's check again. Wait, the numerator of the second fraction is \(6x + 42\), denominator \(x^2 + 14x - 96\). Wait, maybe I made a mistake in factoring. Alternatively, maybe the problem is \(\frac{x^2 - 11x + 24}{3x - 9} \cdot \frac{6x + 42}{x^2 + 14x - 96}\). Let's proceed with factoring what we can:

\(x^2 - 11x + 24=(x - 3)(x - 8)\) (correct, since \((-3)\times(-8)=24\), \(-3 + (-8)=-11\)).
\(3x - 9 = 3(x - 3)\).
\(6x + 42 = 6(x + 7)\).
\(x^2 + 14x - 96\): Let's use quadratic formula: \(x=\frac{-14\pm\sqrt{14^2 - 4\times1\times(-96)}}{2\times1}=\frac{-14\pm\sqrt{196 + 384}}{2}=\frac{-14\pm\sqrt{580}}{2}=\frac{-14\pm2\sqrt{145}}{2}=-7\pm\sqrt{145}\). Hmm, that can't be factored with integers. Wait, maybe the denominator is \(x^2 + 14x - 96\) is correct, but maybe a typo. Alternatively, maybe the denominator is \(x^2 + 14x - 96\) and we proceed.

So substituting factored forms:
\(\frac{(x - 3)(x - 8)}{3(x - 3)} \cdot \frac{6(x + 7)}{x^2 + 14x - 96}\)

Step2: Cancel common factors

Cancel \((x - 3)\) from numerator and denominator of the first fraction:
\(\frac{(x - 8)}{3} \cdot \frac{6(x + 7)}{x^2 + 14x - 96}\)

Simplify \(\frac{6}{3}=2\):
\(\frac{(x - 8) \cdot 2(x + 7)}{x^2 + 14x - 96}\)

Now, check if \(x^2 + 14x - 96\) can be factored with the numerator. Wait, numerator is \(2(x - 8)(x + 7)\), denominator is \(x^2 + 14x - 96\). Let's expand numerator: \(2(x^2 - x - 56)\). Denominator: \(x^2 + 14x - 96\). No common factors. Wait, maybe the denominator was a typo, like \(x^2 + 14x - 56\)? No. Alternatively, maybe the original problem has a different denominator. Wait, maybe the second denominator is \(x^2 + 14x - 96\) and we proceed. So the simplified form is \(\frac{2(x - 8)(x + 7)}{x^2 + 14x - 96}\) or if we made a mistake in factoring the denominator. Wait, let's re-express:

Wait, initial factoring:

\(x^2 - 11x + 24=(x - 3)(x - 8)\) (correct, since \((x - 3)(x - 8)=x^2 - 11x + 24\))
\(3x - 9 = 3(x - 3)\) (correct)
\(6x + 42 = 6(x + 7)\) (correct)
\(x^2 + 14x - 96\): Let's use quadratic formula: \(x=\frac{-14\pm\sqrt{14^2 + 384}}{2}=\frac{-14\pm\sqrt{196 + 384}}{2}=\frac{-14\pm\sqrt{580}}{2}=\frac{-14\pm2\sqrt{145}}{2}=-7\pm\sqrt{145}\). So it doesn't factor with integers. So maybe there's a typo in the problem, but assuming the problem is as written, we proceed with canceling the common \((x - 3)\) and simplifying \(\frac{6}{3}=2\), so:

\(\frac{(x - 8)}{3} \cdot \frac{6(x + 7)}{x^2 + 14x - 96}=\frac{(x - 8) \cdot 2(x + 7)}{x^2 + 14x - 96}=\frac{2(x^2 - x - 56)}{x^2 + 14x - 96}\)

But maybe the intended denominator was \(x^2 + 14x - 56\), but that's speculation. Alternatively, maybe the original problem has a different denominator, like \(x^2 + 14x - 96\) is correct, and we leave it as is.

Wait, maybe I made a mistake in the denominator's sign. If the denominator was \(x^2 - 14x - 96\), then factors would be \((x - 16)(x + 6)\), but that doesn't help. Alternatively, maybe the numerator of the second fraction is \(6x + 42\) and denominator \(x^2 + 14x - 96\) is correct, so the simplified form after canceling \((x - 3)\) and simplifying \(\frac{6}{3}=2\) is \(\frac{2(x - 8)(x + 7)}{x^2 + 14x - 96}\).

Answer:

\(\frac{2(x - 8)(x + 7)}{x^2 + 14x - 96}\) (or simplified further if denominator factors, but as per given, this is the simplified form after canceling common factors)