Find three crossings of an exponential and a cubic

Sketch $y=(e^{\frac54x}+6)-(-2x^3-4x^2+5x+15)$. The source gives three zeros approximately $-2.22,-1.29,1.27$. Its successive signs are negative, positive, negative, positive. The nonpositive portions give the printed result $x\leq-2.22$ or $-1.29\leq x\leq1.27$, with these boundaries understood as rounded readings.

Sketch $y=e^{\frac54x}+6$ and $y=-2x^3-4x^2+5x+15$ on the same axes. The exponential has horizontal asymptote $y=6$. Mark the same three intersection inputs. The exponential is on or below the cubic before the first crossing and between the second and third crossings, giving the same intervals.

Let $\alpha<\beta<\gamma$ be the exact roots of $e^{\frac54x}+2x^3+4x^2-5x-9=0$. Numerically they are $\alpha\approx-2.219106617$, $\beta\approx-1.287564121$ and $\gamma\approx1.265696953$. The exact set is $(-\infty,\alpha]\cup[\beta,\gamma]$.