Solve -3x^2-4x-1

$$-3 { x }^{ 2 } -4x-1$$

Factor

$$-\left(x+1\right)\left(3x+1\right)$$

Solution Steps

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.

$$a+b=-4 ab=-3\left(-1\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-3$$

Rewrite -3x^{2}-4x-1 as \left(-3x^{2}-x\right)+\left(-3x-1\right).

$$\left(-3x^{2}-x\right)+\left(-3x-1\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(3x+1\right)-\left(3x+1\right)$$

Factor out common term 3x+1 by using distributive property.

$$\left(3x+1\right)\left(-x-1\right)$$

Steps Using the Quadratic Formula

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.

$$-3x^{2}-4x-1=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+12\left(-1\right)}}{2\left(-3\right)}$$

Multiply 12 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16-12}}{2\left(-3\right)}$$

Add 16 to -12.

$$x=\frac{-\left(-4\right)±\sqrt{4}}{2\left(-3\right)}$$

Take the square root of 4.

$$x=\frac{-\left(-4\right)±2}{2\left(-3\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±2}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{4±2}{-6}$$

Now solve the equation x=\frac{4±2}{-6} when ± is plus. Add 4 to 2.

$$x=\frac{6}{-6}$$

Divide 6 by -6.

$$x=-1$$

Now solve the equation x=\frac{4±2}{-6} when ± is minus. Subtract 2 from 4.

$$x=\frac{2}{-6}$$

Reduce the fraction \frac{2}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -\frac{1}{3} for x_{2}.

$$-3x^{2}-4x-1=-3\left(x-\left(-1\right)\right)\left(x-\left(-\frac{1}{3}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-3x^{2}-4x-1=-3\left(x+1\right)\left(x+\frac{1}{3}\right)$$

Add \frac{1}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-3x^{2}-4x-1=-3\left(x+1\right)\times \frac{-3x-1}{-3}$$

Cancel out 3, the greatest common factor in -3 and 3.

$$-3x^{2}-4x-1=\left(x+1\right)\left(-3x-1\right)$$

Evaluate

$$-\left(x+1\right)\left(3x+1\right)$$