Find the intersection point/s of the equations x²+7x-3 and 3x+4

Like many things in Mathematics, we have to remember that getting used to visualising the problem is often a fundamental stepping stone towards success. Both equations represent a line on the Cartesian plane and their intersection is to be interpreted as "the point/s of overlap".
In practical terms, we start by writing the following equation out: x²+7x-3 = 3x+4. This way, we are literally stating that the equations are equal to each other. Most likely this won't be true for all values of x, but if this is true for at least one value of x re-arranging the equation to "solve for x" would give us at least the one value of x for which these two equations actually interest, and hence equal each other.
The actual execution of the rearrangement process is just a matter of practice, but this is how I would carry it out... Subtract all common polynomial integers from both sides, to reduce the number of polynomial values you are working with. This should give you: x²+4x-1= 0. At this point, to solve a quadratic equation that equals 0 you have three non-calculator tools available which are a) Factorisation b) "Completing the square" c) The quadratic formula. I would solve this example using "completing the square", which hopefully will soon become part of your repertoire of knowledge if it isn't already. If not, I'd be happy to explain individually each of the three methods listed above.
Once you have successfully rearranged the equation and found the value/s of x for which the graphs intersect, you basically have found the point on the x-plane at which the graphs intersect, but not the points on the y-plane. If you re-insert that value of x into either of the originally stated equations you will find the relevant y-values too. Now you will be able to pin-point exact points of intersection that share both an x and a y value.