To solve this equation, we will first expand the left side by using the distributive property:

(4x - 5)(x + 3) = 4x(x) + 4x(3) - 5(x) - 5(3)
= 4x^2 + 12x - 5x - 15
= 4x^2 + 7x - 15

Now we can set this equal to the right side of the equation and solve for x:

4x^2 + 7x - 15 = 2x - 3

Subtract 2x and add 3 to both sides:

4x^2 + 7x - 2x = 15 + 3
4x^2 + 5x = 18

This equation does not simplify further, so the final answer is:

4x^2 + 5x = 18.