f. \( 5-4 x-x^{2}>0 \)

To solve the inequality \( 5 - 4x - x^2 > 0 \), first rearrange it into standard form: \( -x^2 - 4x + 5 > 0 \). You can flip the inequality by multiplying by -1 (and reversing the sign), yielding \( x^2 + 4x - 5 < 0 \). Next, factor the quadratic to find its roots: \( (x + 5)(x - 1) < 0 \). The roots are \( x = -5 \) and \( x = 1 \). Analyze intervals: between the roots, the expression is negative, so the solution is \( -5 < x < 1 \). Finding the roots of a quadratic function can be tricky. A common mistake is forgetting to check the intervals around the roots. After identifying the critical points, always test points in each interval to confirm whether the original inequality holds. For example, check a value from each interval such as -6, 0, and 2 to validate the sign.