This equation will have one solution when \( h = -4 \) because you get one solution when you have a different number of \( x \)'s on either side of the equation and the coefficients of \( x \) do not match in such a way that allows for infinite solutions or contradictions.
To elaborate, if you let \( h = -4 \), both sides maintain a linear relationship but will intersect at a single point based on the condition imposed by the \( h \) value. This ensures that the equation simplifies in a way to isolate \( x \) effectively leading to one unique solution.
If we re-write the equation \( h(-3x - 6) = -12x - 36 \), we can solve for \( h \):
- Distributing \( h \): \( -3hx - 6h = -12x - 36 \).
- Rearranging for \( x \): Set the coefficients of \( x \) on both sides equal, which gives us \( -3h = -12 \) or \( h = 4 \).
But if we find a scenario where the equation structurally allows both sides to have varying coefficients of \( x \), that justifies a unique solution. Therefore, confirming \( h = -4 \) provides the required conditions for the insights provided.
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