What are the x-intercepts of 4x^ + 8x - 5

Assuming the exponent is supposed to be "^2" your equation will read:

4x² + 8x - 5

It must set equal to y to be a valid function and the y must be set equal to zero to find x-intercepts.

4x² + 8x - 5 = 0

4x² - 2x + 10x - 5 = 0

2x (2x - 1) + 5 (2x - 1) = 0

(2x + 5) (2x - 1) = 0

Set each binomial equal to zero.

2x + 5 = 0

2x = 0 - 5

2x = - 5

Divide both sides by 2

x = - 5/2

2x - 1 = 0

2x = 0 + 1

2x = 1

x = 1/2

Your x-intercepts are x = - 5/2, 1/2 or ( - 5/2, 0) and (1/2, 0)