Solve (11m^2+30m+16)

$$( 11 m ^ { 2 } + 30 m + 16 )$$

Factor

$$\left(m+2\right)\left(11m+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11m^{2}+am+bm+16. To find a and b, set up a system to be solved.

$$a+b=30 ab=11\times 16=176$$

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 176.

$$1,176 2,88 4,44 8,22 11,16$$

Calculate the sum for each pair.

$$1+176=177 2+88=90 4+44=48 8+22=30 11+16=27$$

The solution is the pair that gives sum 30.

$$a=8 b=22$$

Rewrite 11m^{2}+30m+16 as \left(11m^{2}+8m\right)+\left(22m+16\right).

$$\left(11m^{2}+8m\right)+\left(22m+16\right)$$

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

$$m\left(11m+8\right)+2\left(11m+8\right)$$

Factor out common term 11m+8 by using distributive property.

$$\left(11m+8\right)\left(m+2\right)$$

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.

$$11m^{2}+30m+16=0$$

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.

$$m=\frac{-30±\sqrt{30^{2}-4\times 11\times 16}}{2\times 11}$$

Square 30.

$$m=\frac{-30±\sqrt{900-4\times 11\times 16}}{2\times 11}$$

Multiply -4 times 11.

$$m=\frac{-30±\sqrt{900-44\times 16}}{2\times 11}$$

Multiply -44 times 16.

$$m=\frac{-30±\sqrt{900-704}}{2\times 11}$$

Add 900 to -704.

$$m=\frac{-30±\sqrt{196}}{2\times 11}$$

Take the square root of 196.

$$m=\frac{-30±14}{2\times 11}$$

Multiply 2 times 11.

$$m=\frac{-30±14}{22}$$

Now solve the equation m=\frac{-30±14}{22} when ± is plus. Add -30 to 14.

$$m=-\frac{16}{22}$$

Reduce the fraction \frac{-16}{22} to lowest terms by extracting and canceling out 2.

$$m=-\frac{8}{11}$$

Now solve the equation m=\frac{-30±14}{22} when ± is minus. Subtract 14 from -30.

$$m=-\frac{44}{22}$$

Divide -44 by 22.

$$m=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{8}{11} for x_{1} and -2 for x_{2}.

$$11m^{2}+30m+16=11\left(m-\left(-\frac{8}{11}\right)\right)\left(m-\left(-2\right)\right)$$

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

$$11m^{2}+30m+16=11\left(m+\frac{8}{11}\right)\left(m+2\right)$$

Add \frac{8}{11} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$11m^{2}+30m+16=11\times \frac{11m+8}{11}\left(m+2\right)$$

Cancel out 11, the greatest common factor in 11 and 11.

$$11m^{2}+30m+16=\left(11m+8\right)\left(m+2\right)$$

Evaluate

$$\left(m+2\right)\left(11m+8\right)$$