To solve for the width \( w \) of the trampoline using the equation \( w^2 + 6w - 112 = 0 \), we can either factor the quadratic equation or use the quadratic formula. In this case, let's try to factor it.
We need to find two numbers that multiply to \(-112\) (the constant term) and add to \(6\) (the coefficient of \(w\)).
The factors of \(-112\) that satisfy this are \(14\) and \(-8\), since: \[ 14 \times (-8) = -112 \quad \text{and} \quad 14 + (-8) = 6. \]
Thus, we can factor the quadratic as: \[ (w + 14)(w - 8) = 0. \]
Setting each factor equal to zero gives us: \[ w + 14 = 0 \quad \Rightarrow \quad w = -14 \quad \text{(not a valid width)}, \] \[ w - 8 = 0 \quad \Rightarrow \quad w = 8. \]
The only valid solution for the width of the trampoline is: \[ \boxed{8} \text{ feet}. \]
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