Solve 4m^2+8m+3

$$4 { m }^{ 2 } +8m+3$$

Factor

$$\left(2m+1\right)\left(2m+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4m^{2}+am+bm+3. To find a and b, set up a system to be solved.

$$a+b=8 ab=4\times 3=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=2 b=6$$

Rewrite 4m^{2}+8m+3 as \left(4m^{2}+2m\right)+\left(6m+3\right).

$$\left(4m^{2}+2m\right)+\left(6m+3\right)$$

Factor out 2m in the first and 3 in the second group.

$$2m\left(2m+1\right)+3\left(2m+1\right)$$

Factor out common term 2m+1 by using distributive property.

$$\left(2m+1\right)\left(2m+3\right)$$

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.

$$4m^{2}+8m+3=0$$

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.

$$m=\frac{-8±\sqrt{8^{2}-4\times 4\times 3}}{2\times 4}$$

Square 8.

$$m=\frac{-8±\sqrt{64-4\times 4\times 3}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-8±\sqrt{64-16\times 3}}{2\times 4}$$

Multiply -16 times 3.

$$m=\frac{-8±\sqrt{64-48}}{2\times 4}$$

Add 64 to -48.

$$m=\frac{-8±\sqrt{16}}{2\times 4}$$

Take the square root of 16.

$$m=\frac{-8±4}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{-8±4}{8}$$

Now solve the equation m=\frac{-8±4}{8} when ± is plus. Add -8 to 4.

$$m=-\frac{4}{8}$$

Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

$$m=-\frac{1}{2}$$

Now solve the equation m=\frac{-8±4}{8} when ± is minus. Subtract 4 from -8.

$$m=-\frac{12}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

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

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

$$4m^{2}+8m+3=4\left(m-\left(-\frac{1}{2}\right)\right)\left(m-\left(-\frac{3}{2}\right)\right)$$

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

$$4m^{2}+8m+3=4\left(m+\frac{1}{2}\right)\left(m+\frac{3}{2}\right)$$

Add \frac{1}{2} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4m^{2}+8m+3=4\times \frac{2m+1}{2}\left(m+\frac{3}{2}\right)$$

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

$$4m^{2}+8m+3=4\times \frac{2m+1}{2}\times \frac{2m+3}{2}$$

Multiply \frac{2m+1}{2} times \frac{2m+3}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$4m^{2}+8m+3=4\times \frac{\left(2m+1\right)\left(2m+3\right)}{2\times 2}$$

Multiply 2 times 2.

$$4m^{2}+8m+3=4\times \frac{\left(2m+1\right)\left(2m+3\right)}{4}$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4m^{2}+8m+3=\left(2m+1\right)\left(2m+3\right)$$

Evaluate

$$\left(2m+1\right)\left(2m+3\right)$$