To find the values for which the rational expression \(\frac{2x+4}{x^2+6x+8}\) is undefined, we need to determine when the denominator is equal to zero. The expression is undefined wherever the denominator is zero.
First, let's set the denominator \(x^2 + 6x + 8\) equal to zero:
\[ x^2 + 6x + 8 = 0 \]
Next, we will factor this quadratic equation. We need to find two numbers that multiply to \(8\) (the constant term) and add up to \(6\) (the coefficient of the linear term). The numbers \(2\) and \(4\) fit this requirement, since:
\[ 2 \cdot 4 = 8 \quad \text{and} \quad 2 + 4 = 6 \]
Thus, we can factor the quadratic as follows:
\[ (x + 2)(x + 4) = 0 \]
To find the values of \(x\) that make the denominator zero, we set each factor equal to zero:
- \(x + 2 = 0\) → \(x = -2\)
- \(x + 4 = 0\) → \(x = -4\)
Therefore, the rational expression \(\frac{2x+4}{x^2+6x+8}\) is undefined for:
\[ x = -2 \quad \text{and} \quad x = -4 \]
Thus, the set of numbers for which the rational expression is undefined is \({-2, -4}\).