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a+b=-3 ab=-28=-28
Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+28. To find a and b, set up a system to be solved.
1,-28 2,-14 4,-7
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -28.
1-28=-27 2-14=-12 4-7=-3
Calculate the sum for each pair.
a=4 b=-7
The solution is the pair that gives sum -3.
\left(-x^{2}+4x\right)+\left(-7x+28\right)
Rewrite -x^{2}-3x+28 as \left(-x^{2}+4x\right)+\left(-7x+28\right).
x\left(-x+4\right)+7\left(-x+4\right)
Factor out x in the first and 7 in the second group.
\left(-x+4\right)\left(x+7\right)
Factor out common term -x+4 by using distributive property.
-x^{2}-3x+28=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 28}}{2\left(-1\right)}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 28}}{2\left(-1\right)}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9+4\times 28}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-3\right)±\sqrt{9+112}}{2\left(-1\right)}
Multiply 4 times 28.
x=\frac{-\left(-3\right)±\sqrt{121}}{2\left(-1\right)}
Add 9 to 112.
x=\frac{-\left(-3\right)±11}{2\left(-1\right)}
Take the square root of 121.
x=\frac{3±11}{2\left(-1\right)}
The opposite of -3 is 3.
x=\frac{3±11}{-2}
Multiply 2 times -1.
x=\frac{14}{-2}
Now solve the equation x=\frac{3±11}{-2} when ± is plus. Add 3 to 11.
x=-7
Divide 14 by -2.
x=\frac{-8}{-2}
Now solve the equation x=\frac{3±11}{-2} when ± is minus. Subtract 11 from 3.
x=4
Divide -8 by -2.
-x^{2}-3x+28=-\left(x-\left(-7\right)\right)\left(x-4\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and 4 for x_{2}.
-x^{2}-3x+28=-\left(x+7\right)\left(x-4\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.