Factorise 4x2+8x+3 completely.

To factorise the quadratic equation 4x^2 + 8x + 3, we first need to multiply the coefficient of x^2 (which is 4) and the constant term (which is 3). This gives us 12.

Next, we need to find two numbers that multiply to give 12 and add to give 8 (the coefficient of x). The numbers that satisfy these conditions are 6 and 2.

We then rewrite the middle term of the quadratic equation (8x) as the sum of 6x and 2x. This gives us:

4x^2 + 6x + 2x + 3

Next, we factor by grouping. The first two terms have a common factor of 2x, and the last two terms have a common factor of 1. This gives us:

2x(2x + 3) + 1(2x + 3)

Finally, we notice that (2x + 3) is a common factor, so we can factor it out to give the final factorised form of the quadratic equation:

(2x + 3)(2x + 1)

So, 4x^2 + 8x + 3 factorises to (2x + 3)(2x + 1).