Question

which expression is equivalent to the expression below? \\(\frac{\frac{\frac{8c^2 + 3c}{-4c + 2}}{2c + 1}}{4c - 3}\\)

Explanation:

Step1: Simplify the numerator

First, we have the numerator as \(8c^{2}+3c - 4c + 2\). Combine like terms: \(3c-4c=-c\), so the numerator becomes \(8c^{2}-c + 2\). Wait, maybe I misread the original expression. Wait, the original expression seems to be a fraction with multiple layers? Wait, looking at the image, maybe the expression is \(\frac{\frac{\frac{8c^{2}+3c}{-4c + 2}}{2c + 1}}{4c-3}\)? Wait, maybe the correct original expression is a complex fraction: \(\frac{\frac{8c^{2}+3c}{-4c + 2}}{2c + 1}\div(4c - 3)\) or \(\frac{\frac{8c^{2}+3c}{-4c + 2}}{(2c + 1)(4c - 3)}\)? Wait, maybe there is a typo, but let's assume the expression is \(\frac{\frac{8c^{2}+3c}{-4c + 2}}{2c + 1}\) divided by \(4c - 3\), which is \(\frac{8c^{2}+3c}{(-4c + 2)(2c + 1)(4c - 3)}\). Wait, no, maybe the original expression is \(\frac{8c^{2}+3c}{-4c + 2}\div(2c + 1)\div(4c - 3)\), which is \(\frac{8c^{2}+3c}{(-4c + 2)(2c + 1)(4c - 3)}\). Wait, maybe factor numerator and denominator.

Wait, maybe the correct approach is to factor the numerator \(8c^{2}+3c\). Let's factor \(c\): \(c(8c + 3)\). Denominator: \(-4c + 2=-2(2c - 1)\), then we have \(\frac{c(8c + 3)}{-2(2c - 1)}\div(2c + 1)\div(4c - 3)\). Dividing by a number is multiplying by its reciprocal, so \(\frac{c(8c + 3)}{-2(2c - 1)}\times\frac{1}{2c + 1}\times\frac{1}{4c - 3}\). Wait, this seems complicated. Maybe I misread the expression. Alternatively, maybe the original expression is \(\frac{8c^{2}+3c}{-4c + 2}\) divided by \((2c + 1)(4c - 3)\), so \(\frac{8c^{2}+3c}{(-4c + 2)(2c + 1)(4c - 3)}\). Let's check if numerator and denominator can be factored. Numerator: \(8c^{2}+3c = c(8c + 3)\). Denominator: \(-4c + 2=-2(2c - 1)\), \(2c + 1\) is prime, \(4c - 3\) is prime. So no common factors. Wait, maybe the original expression is different. Wait, maybe the user made a typo, but assuming the expression is \(\frac{8c^{2}+3c}{-4c + 2}\) divided by \((2c + 1)\) and then divided by \((4c - 3)\), the simplified form would be \(\frac{c(8c + 3)}{-2(2c - 1)(2c + 1)(4c - 3)}\). But maybe I misread the expression. Alternatively, maybe the original expression is \(\frac{8c^{2}+3c}{-4c + 2}\) over \((2c + 1)\) over \((4c - 3)\), which is \(\frac{8c^{2}+3c}{(-4c + 2)(2c + 1)(4c - 3)}\).

Wait, maybe the correct expression is \(\frac{8c^{2}+3c}{-4c + 2}\div(2c + 1)\div(4c - 3)\), so:

Step1: Rewrite division as multiplication

\(\frac{8c^{2}+3c}{-4c + 2}\times\frac{1}{2c + 1}\times\frac{1}{4c - 3}\)

Step2: Factor numerator and denominator

Numerator: \(8c^{2}+3c = c(8c + 3)\)

Denominator: \(-4c + 2=-2(2c - 1)\), so denominator becomes \(-2(2c - 1)(2c + 1)(4c - 3)\)

So the expression is \(\frac{c(8c + 3)}{-2(2c - 1)(2c + 1)(4c - 3)}\)

But maybe the original expression was different. Wait, maybe the user intended a different expression. Alternatively, maybe the expression is \(\frac{8c^{2}+3c}{-4c + 2}\) divided by \((2c + 1)\) and then multiplied by \((4c - 3)\)? No, the question is "Which expression is equivalent...", so maybe the options are given, but since the user didn't provide options, maybe there was a mistake. Wait, maybe the original expression is \(\frac{8c^{2}+3c}{-4c + 2}\div(2c + 1)\div(4c - 3)\), and the equivalent expression is \(\frac{c(8c + 3)}{-2(2c - 1)(2c + 1)(4c - 3)}\), but without options, it's hard to tell. Wait, maybe I misread the original expression. Let's re-express the image: the vertical lines are division, so the expression is \(\frac{\frac{\frac{8c^{2}+3c}{-4c + 2}}{2c + 1}}{4c - 3}\), which is \(\frac{8c^{2}+3c}{(-4c + 2)(2c + 1)(4c - 3)}\). Let's simplify numerator and denominator:

Numerator: \(8c^{2}+3c = c(8c + 3)\)

Denominator: \(-4c + 2=-2(2c - 1)\), so denominator is \(-2(2c - 1)(2c + 1)(4c - 3)\)

So the expression is \(\frac{c(8c + 3)}{-2(2c - 1)(2c + 1)(4c - 3)}\), which can be written as \(-\frac{c(8c + 3)}{2(2c - 1)(2c + 1)(4c - 3)}\)

But since the user didn't provide options, maybe there was a mistake in the input. Alternatively, maybe the original expression is \(\frac{8c^{2}+3c}{-4c + 2}\) divided by \((2c + 1)\) and then divided by \((4c - 3)\), and the equivalent expression is \(\frac{c(8c + 3)}{-2(2c - 1)(2c + 1)(4c - 3)}\)

Answer:

Assuming the expression is \(\frac{\frac{\frac{8c^{2}+3c}{-4c + 2}}{2c + 1}}{4c - 3}\), the equivalent expression is \(\boldsymbol{-\frac{c(8c + 3)}{2(2c - 1)(2c + 1)(4c - 3)}}\) (or simplified form depending on options, but since options are not provided, this is the simplified form).