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a+b=1 ab=4\left(-33\right)=-132
Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-33. To find a and b, set up a system to be solved.
-1,132 -2,66 -3,44 -4,33 -6,22 -11,12
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -132.
-1+132=131 -2+66=64 -3+44=41 -4+33=29 -6+22=16 -11+12=1
Calculate the sum for each pair.
a=-11 b=12
The solution is the pair that gives sum 1.
\left(4x^{2}-11x\right)+\left(12x-33\right)
Rewrite 4x^{2}+x-33 as \left(4x^{2}-11x\right)+\left(12x-33\right).
x\left(4x-11\right)+3\left(4x-11\right)
Factor out x in the first and 3 in the second group.
\left(4x-11\right)\left(x+3\right)
Factor out common term 4x-11 by using distributive property.
4x^{2}+x-33=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{-1±\sqrt{1^{2}-4\times 4\left(-33\right)}}{2\times 4}
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{-1±\sqrt{1-4\times 4\left(-33\right)}}{2\times 4}
Square 1.
x=\frac{-1±\sqrt{1-16\left(-33\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-1±\sqrt{1+528}}{2\times 4}
Multiply -16 times -33.
x=\frac{-1±\sqrt{529}}{2\times 4}
Add 1 to 528.
x=\frac{-1±23}{2\times 4}
Take the square root of 529.
x=\frac{-1±23}{8}
Multiply 2 times 4.
x=\frac{22}{8}
Now solve the equation x=\frac{-1±23}{8} when ± is plus. Add -1 to 23.
x=\frac{11}{4}
Reduce the fraction \frac{22}{8} to lowest terms by extracting and canceling out 2.
x=-\frac{24}{8}
Now solve the equation x=\frac{-1±23}{8} when ± is minus. Subtract 23 from -1.
x=-3
Divide -24 by 8.
4x^{2}+x-33=4\left(x-\frac{11}{4}\right)\left(x-\left(-3\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11}{4} for x_{1} and -3 for x_{2}.
4x^{2}+x-33=4\left(x-\frac{11}{4}\right)\left(x+3\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
4x^{2}+x-33=4\times \frac{4x-11}{4}\left(x+3\right)
Subtract \frac{11}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
4x^{2}+x-33=\left(4x-11\right)\left(x+3\right)
Cancel out 4, the greatest common factor in 4 and 4.