jmc

We look at the coefficient of $x$ in the expansion of the product on the left. We get an $x$ term when we multiply $(+7)(+tx)$ and when we multiply $(-6x)(+10)$ in the expansion. So, on the left the $x$ term is $7tx-60x$. Since this term must equal $-102x$, we have $7tx-60x=-102x$, so $t=\boxed{-6}$.

We can check our answer (and check that it is indeed possible to find a solution to this problem) by multiplying out the left when $t=-6$: $$\begin{array}{rl}& (5x^2-6x+7)(4x^2-6x+10)\\ & =5x^2(4x^2-6x+10)-6x(4x^2-6x+10)\\ & +7(4x^2-6x+10)\\ & =20x^4-54x^3+114x^2-102x+70.\end{array}$$This matches the polynomial given in the problem, so our answer is correct.