4x^2 - 8x - 5
Ok so I'll try to explain this the way I understand it. Usually to factor you have to find 2 numbers that multiply to equal the last number, and add up to equal the number part of the middle. I'll just use \[ax ^{2}+bx+c\] as an example equation. So you have to find 2 numbers that multiply to get c, and add up to get b. Then you would usually take those 2 numbers and use it to factor and get (x+?)(x+?). But since there is a number for a that is not 1 in your equation, it's a little different.
First, multiply the first and last numbers 4 and -5 (a and c in the example equation). You get -20. So try to find 2 number that are factors of -20 and add up to equal 8. Just start to list the factors off on paper or in your head. Remember to include the negatives.
-1, 20
1, -20
-2, 10
2, -10 These add up to equal -8
Then put 2 and -10 to separate the middle part of the equation. \[4x ^{2}+2x-10x-5\]Then group the equation into 2 separate parts. \[(4x ^{2}+2x)+(-10x-5)\]Then you take out any common factors from the parentheses.\[2x(2x+1)-5(2x+1)\]And then group together the parts you took out, which are the 2x and -5.\[(2x-5)(2x+1)\]It becomes this because 2x+1 are the same.
There's your answer. Hope this helps!