Final answer:
The given trinomial cannot be factored using rational coefficients because there are no two numbers that multiply to 4 and add up to 3. Instead, the quadratic formula is used to find the solutions: v=1/2 and v=-5/4.
Step-by-step explanation:
To factor the given trinomial, we first write it in the standard form. Therefore, rearrange the equation 4v^(2)+3v-5=0.
When factoring trinomials, we look for two numbers that multiply to the coefficient of the v^2 term (which is 4 in this case) and sum to the coefficient of the v term (which is 3). Unfortunately, no such pair of numbers exist in this case, which implies the trinomial does not factorize over the integers. Therefore, this trinomial can't be factored using rational coefficients, and the solutions of the equation are irrational.
In order to find these solutions, you could use the quadratic formula (-b±sqrt(b^2-4ac))/2a. For this equation, a=4, b=3, c=-5. Plugging these values into the formula we get:
(-3±sqrt((3)^2-4*4*(-5)))/2*4, which simplifies to v=(-3±sqrt(49))/8.
Thus the solutions of the equation are v=(-3+7)/8=1/2 and v=(-3-7)/8=-5/4.
Learn more about Factoring Trinomials