In order for this polynomial (with respect to x) to have real roots, its discriminant should be non negative.
[tex]9 {m}^{2} - (4 \times 3 \times ( {m}^{2} - m - 3) \geqslant 0[/tex]
Which means
[tex]9 {m}^{2} - 12 {m}^{2} + 12m + 36 \geqslant 0[/tex]
[tex] - 3 {m}^{2} + 12m + 36 \geqslant 0[/tex]
[tex] {m}^{2} - 4m - 12 \leqslant 0[/tex]
The last inequality is also a polynomial inequality.
The polynomial (with respect to m) has the following discriminant:
[tex] {4}^{2} - 4 \times - 1 \times 12 \times 1 = 16 + 48 = 64[/tex]
Thus the roots of this polynomial are:
[tex] \frac{4( + - )8}{2} = 6 \: and \: - 1[/tex]
When a polynomial has two positive roots named y, z for example where y<z the sign of the polynomial goes as following.
If the coefficient of the highest order term is positive then the polynomial is positive for m<y and m>z and negative for y<m<z.
If the coefficient of the highest order term is negative then the polynomial is negative for m<y and m>z and positive for y<m<z.
In the polynomial:
[tex] {m}^{2} - 4m - 12[/tex]
The coefficient of the highest order term is 1 thus the polynomial is non positive at the interval [-1,6].
Thus the solution to the exercise is
[tex] - 1 \leqslant m \leqslant 6[/tex]
I apologize for any typos or wrong calculations.
