Factoring $x^2 + 4x -3$ step by step.

Question

I have the following equation that I need to factor: $x^2 + 4x -3$. I cannot use the factoring by grouping method as there are no integers that add up to $4$ and give $-3$ when multiplied.
What method should I use?
Could you please give me a step by step explanation?

Answer 1

$x^2 + 4x - 3 = (x+2)^2 - 7 = 0 \iff (x+2)^2 = 7 \iff x+2 = \pm\sqrt{7} \iff x = -2 \pm \sqrt{7}\implies x^2 + 4x - 3 = (x - (-2-\sqrt{7}))(x-(-2+\sqrt{7})) = (x+(2+\sqrt{7}))(x+(2-\sqrt{7})).$

Answer 2

This is a shitty one:
$x^2+3x-4=0$ can easily be solved as $x=1$ and $x=-4$, so you guess $x^2+4x-3$ should be similar.

Sorry, it is not!

Here's another approach (next to the general discriminant method):

$x^2+4x-3 = x^2+4x+4-7 = (x+2)^2-\sqrt{7}^2 = (x+2+\sqrt{7}) \cdot (x+2-\sqrt{7})$

A piece of advise: only do this with equations of the form $x^2 \pm ex ... = 0$, where $e$ is any even number :-)

Answer 3

$$ x^2+4x-3=x^2-4x+4-7=(x-2)^2-(\sqrt{7})^2=(x-2-\sqrt{7})(x-2+\sqrt{7}) $$