What is the sum of the $x$-values that satisfy the equation $5=\frac{x^3-2x^2-8x}{x+2}$?
We can factor $x$ out of the numerator so that we're left with $$\frac{x(x^2-2x-8)}{x+2}=\frac{x(x-4)(x+2)}{x+2}$$After canceling out the $x+2$ from the numerator and denominator, we have $x(x-4)=5$.  Solving for the roots of a quadratic equation, we have $x^2-4x-5=0$, which gives us $(x-5)(x+1)=0$ and $x=5$ or $x=-1$. The sum of these values is $\boxed{4}$, which is our answer.

Alternatively, since the sum of the solutions for a quadratic with the equation $ax^2+bx+c=0$ is $-b/a$, the sum of the zeros of the quadratic $x^2-4x-5$ is $4/1=\boxed{4}$.
4
Level 5
Algebra
Problem: What is the sum of the $x$-values that satisfy the equation $5=\frac{x^3-2x^2-8x}{x+2}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation, first, we need to eliminate the fraction by multiplying both sides by \(x + 2\), assuming \(x \neq -2\). This gives us the equation \(5(x + 2) = x^3 - 2x^2 - 8x\). Expanding the left side results in \(5x + 10 = x^3 - 2x^2 - 8x\). 

Next, we rearrange the equation to set it to zero: \(x^3 - 2x^2 - 8x - 5x - 10 = 0\), which simplifies to \(x^3 - 2x^2 - 13x - 10 = 0\). 

To find the roots of this cubic equation, we can use the Rational Root Theorem to test possible rational roots. Testing \(x = -1\) yields \((-1)^3 - 2(-1)^2 - 13(-1) - 10 = -1 - 2 + 13 - 10 = 0\), confirming that \(x = -1\) is a root.

Next, we perform synthetic division of \(x^3 - 2x^2 - 13x - 10\) by \(x + 1\) to factor the cubic polynomial. The result of the synthetic division is \(x^2 - 3x - 10\). 

Now, we need to factor \(x^2 - 3x - 10\). This factors to \((x - 5)(x + 2) = 0\). Thus, the complete factorization of the original cubic equation is \((x + 1)(x - 5)(x + 2) = 0\).

The solutions to this equation are \(x = -1\), \(x = 5\), and \(x = -2\). However, we must discard \(x = -2\) because it makes the original equation undefined.

The remaining valid solutions are \(x = -1\) and \(x = 5\). To find the sum of the \(x\)-values that satisfy the equation, we calculate \(-1 + 5 = 4\).

Thus, the final answer is \(4\).<|END_OF_LATENT|> The final answer is $4$.