Find if the expansion of the product of and has no term.

Find  if the expansion of the product of  and  has no  term.

$$\small{\text{ $ (x^3 - 4x^2 + 2x - 5)*( x^2 + tx - 7) = tx^4-4tx^3+\textcolor[rgb]{1,0,0}{2tx^2}-5tx+x^5-4x^4-5x^3+\textcolor[rgb]{1,0,0}{23x^2}-14x+35 $ }}$$

$$\\ \small{\text{ Set $ \textcolor[rgb]{1,0,0}{2tx^2}+\textcolor[rgb]{1,0,0}{23x^2} = 0 $ than the product has no $x^2$, t must be a constant! }} \\\\ 2tx^2 + 23x^2 = 0 \\\\ 2tx^2 = - 23x^2 \quad | \quad : 2x^2 \\\\ t= -\frac{23}{2} = - 11.5$$

$$\small{\text{ $ \textcolor[rgb]{1,0,0}{t= -11.5}\qquad (x^3-4x^2+2x-5)*(x^2+\textcolor[rgb]{1,0,0}{(-11.5)}x-7) = x^5 - 15.5x^4+41x^3+43.5x+35 $ }}$$

There is no more $$x^2$$

Nice, heureka......the solution is easy.....figuring out how to get there is the hard part....!!!!!