Solve (3x+1(2x^2))

$$(3x+1(2 { x }^{ 2 } ))$$

Factor

$$x\left(2x+3\right)$$

Solution Steps

Factor out x.

$$x\left(3+2x\right)$$

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.

$$2x^{2}+3x=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}}}{2\times 2}$$

Take the square root of 3^{2}.

$$x=\frac{-3±3}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-3±3}{4}$$

Now solve the equation x=\frac{-3±3}{4} when ± is plus. Add -3 to 3.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

Now solve the equation x=\frac{-3±3}{4} when ± is minus. Subtract 3 from -3.

$$x=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -\frac{3}{2} for x_{2}.

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

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

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

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}+3x=2x\times \frac{2x+3}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

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

Evaluate

$$x\left(2x+3\right)$$

Solution Steps

Multiply 1 and 2 to get 2.

$$3x+2x^{2}$$