If the roots of the equation are at x= -4 and x=3, then we can work backwards to see what equation those roots were derived from. If we factored a quadratic equation and obtained the given solutions, it would mean the factored form looked something like:

Because this is the form that would yield the solutions x= -4 and x=3. If we work backwards and multiply the factors back together, we get the following quadratic equation:

We can make a quadratic polynomial with  by mutiplying the linear polynomials they are roots of, and multiplying them out.

Start

Distribute the negative sign

FOIL the two polynomials. This means multiply the firsts, then the outers, followed by the inners and lastly, the last terms.

Simplify

Since we know the solutions of the equation, we know that:

We simply carry out the multiplication on the left side of the equation to get the quadratic equation.

These two points tell us that the quadratic function has zeros at , and at .

These correspond to the linear expressions , and .

Expand their product and you arrive at the correct answer.

If the quadratic is opening up the coefficient infront of the squared term will be positive. Thus we get:

.

If the quadratic is opening down it would pass through the same two points but have the equation:

.

Since only  is seen in the answer choices, it is the correct answer.

When we solve quadratic equations we get solutions called roots or places where that function crosses the x axis. Not all all will cross the x axis, since we have seen that functions can be shifted around, but many will. When they do this is a special and telling circumstance in mathematics. Since we know that roots of these types of equations are of the form x-k, when given a list of roots we can work backwards to find the equation they pertain to and we do this by multiplying the factors (the foil method).

If you were given an answer of the form  then just foil or multiply the two factors. If you were given only two x values of the roots then put them into the form   that would give you those two x values (when set equal to zero) and multiply to see if you get the original function. 

For our problem the correct answer is .

Use the foil method to get the original quadratic.

First multiply 2x by all terms in  :  

then multiply 2 by all terms in  : .

We then combine    for the final answer. 

The roots or solutions of a quadratic equation are its factors set equal to zero and then solved for x. When roots are given and the quadratic equation is sought, write the roots with the correct sign to give you that root when it is set equal to zero and solved. For example, a quadratic equation has a root of -5 and +3. How could you get that same root if it was set equal to zero? With  and  because they solve to give -5 and +3. Thus, these factors, when multiplied together, will give you the correct quadratic equation.

Our roots were .

So our factors are  and .

Now FOIL these two factors:

First: 

Outer: 

Inner: 

Last: 

Simplify:

Therefore...

These two terms give you the solution. SO, work backwards.

FOIL (Distribute the first term to the second term).

Combine like terms:

When solving a quadratic equation, the first thing to look for is whether or not it can be factored, as this is most often the easiest and fastest method if the quadratic can in fact be factored. We can see that each of the terms in the given equation have a common factor of 3, so it will be easier to factor the quadratic if we first factor out the 3:

Now we're left with a polynomial where we need to find two numbers whose product is -28 and whose sum is -3. Thinking about the factors of 28, we can see that 4 and 7 will yield -3 if 7 is negative and 4 is positive, so we now have our factorization:

To find the roots of an equation in the form , you use the quadratic formula 

.

In our case, we have . 

This gives us  which simplifies to 

Factorize  and set this equation equal to zero.

The answer  is one of the possible choices.