How do you find the intersection of a straight line and a curve?

Question : You have been given the following straight line equation --> y = 2x + 3 and the equation for a curve --> y = x² + 3x + 1.
At the point(s) where these lines intersect we know that the coordinates (x,y) are the same and satisfy both equations. Once we have this clear in our minds we see that we can set them equal to each other as follows as they are both equal to y so therefore equal to each other --> 2x + 3 = x² + 3x + 1.Once this equation has been set up we can now begin to solve for x. We do this by subtracting (2x + 3) from both sides of the equation --> 2x + 3 - 2x - 3 = x² + 3x + 1 -2x - 3.This then simplifies to give --> 0 = x² + x - 2.
We must now factor this equation. We know that it will factor to the form (x + _)(x + _) but we must figure out what numbers will fill the gaps. Imagine that there is a number 1 in front of the x in the equation x² + x - 2 = 0. This means that the 2 numbers in the brackets must add to 1 and also multiply to give - 2. For 2 numbers to multiply and give a negative number there must be one positive and one negative. After some trial and error we find that the 2 numbers that satisfy these sums are -1 and 2 giving (x - 1)(x + 2)=0. You can expand this equation to double check if you have factored correctly. For (x - 1)(x + 2) = 0 to be true at least one of the brackets must equal 0 as anything multiplied by 0 is 0. From this we get our x values to be x = 1 and x = -2.
To determine the y coordinates to match the x values we put the x values in one of the original equations given to us. After doing this we can present our answer by saying there are 2 points of intersection at (1,5) and (-2,-1).