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a+b=2 ab=3\left(-1\right)=-3
Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-1. To find a and b, set up a system to be solved.
a=-1 b=3
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. The only such pair is the system solution.
\left(3x^{2}-x\right)+\left(3x-1\right)
Rewrite 3x^{2}+2x-1 as \left(3x^{2}-x\right)+\left(3x-1\right).
x\left(3x-1\right)+3x-1
Factor out x in 3x^{2}-x.
\left(3x-1\right)\left(x+1\right)
Factor out common term 3x-1 by using distributive property.
3x^{2}+2x-1=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{-2±\sqrt{2^{2}-4\times 3\left(-1\right)}}{2\times 3}
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{-2±\sqrt{4-4\times 3\left(-1\right)}}{2\times 3}
Square 2.
x=\frac{-2±\sqrt{4-12\left(-1\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-2±\sqrt{4+12}}{2\times 3}
Multiply -12 times -1.
x=\frac{-2±\sqrt{16}}{2\times 3}
Add 4 to 12.
x=\frac{-2±4}{2\times 3}
Take the square root of 16.
x=\frac{-2±4}{6}
Multiply 2 times 3.
x=\frac{2}{6}
Now solve the equation x=\frac{-2±4}{6} when ± is plus. Add -2 to 4.
x=\frac{1}{3}
Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.
x=-\frac{6}{6}
Now solve the equation x=\frac{-2±4}{6} when ± is minus. Subtract 4 from -2.
x=-1
Divide -6 by 6.
3x^{2}+2x-1=3\left(x-\frac{1}{3}\right)\left(x-\left(-1\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{3} for x_{1} and -1 for x_{2}.
3x^{2}+2x-1=3\left(x-\frac{1}{3}\right)\left(x+1\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
3x^{2}+2x-1=3\times \frac{3x-1}{3}\left(x+1\right)
Subtract \frac{1}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
3x^{2}+2x-1=\left(3x-1\right)\left(x+1\right)
Cancel out 3, the greatest common factor in 3 and 3.