Find $t$ if the expansion of the product of $x^3 - 4x^2 + 2x - 5$ and $x^2 + tx - 7$ has no $x^2$ term.

The expansion of the product of the given polynomials can be determined through the distributive property of multiplication. 

   (x³ - 4x² + 2x - 5)(x² + tx - 7)

This goes with the first term of the first expression first up until every term of the first expression is multiplied to the terms of the second expression.

 x⁵ + tx⁴ - 7x³ - 4x⁴ - 4tx³ + 28x² + 2x³ + 2tx² - 14x - 5x² - 5tx + 35

The terms with x² are 28x², 2tx², and -5x². The sum of the numerical coefficients should be zero. 
   28 + 2t - 5 = 0

The value of t from the equation is -11.5.