Final answer:
By comparing the coefficients after multiplication, the polynomial equation is matched to determine that the value of k is -10.
Step-by-step explanation:
To find the value of k, we need to compare the coefficients of the given polynomial equations.
The given equation is (x^2 - 2x - 5) f(x) = 2x^4 - 4x^3 + kx^2. We know that f(x) is also a polynomial, so when multiplied by the quadratic (x^2 - 2x - 5), the highest degree term x^2 from (x^2 - 2x - 5) will multiply by the highest degree term from f(x) to give us the x^4 term.
Since the x^4 coefficient is 2 on the right-hand side, the highest degree term in f(x) must be 2x^2 to just get an x^4 after multiplication. Now, this means x^2 times 2x^2 gives us 2x^4, and also, -2x times 2x^2 gives us -4x^3.
Finally, for the x^2 term, -5 times 2x^2 gives us -10x^2.
By comparing it to the given equation, 2x^4 - 4x^3 + kx^2, we can see that k must be equal to -10 since that's the coefficient of the x^2 term on the right-hand side. Hence, the value of k is -10.