Answer:
-23/2 or -11.5
Explanation:
We find the expansion of
(x^3 - 4x^2 + 2x - 5)(x^2 + tx - 7)
by multiplying each term in x^3 - 4x^2 + 2x - 5 by each term in x^2 + tx - 7, and then summing the resulting products. We only care about the x^2 term, so we focus solely on the products that produce an x^2 term. Thus, the x^2 term in the product is equal to
(-4x^2)(-7) + (2x)(tx) + (-5)(x^2) = 28x^2 + 2tx^2 - 5x^2 = (2t + 23)x^2.
Hence, if the product has no x^2 term, then the coefficient of x^2, namely 2t + 23, must be equal to 0. This occurs when t = -23/5.