Solve 21+22m+m^2

$$21 + 22 m + m ^ { 2 }$$

Factor

$$\left(m+1\right)\left(m+21\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$m^{2}+22m+21$$

Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm+21. To find a and b, set up a system to be solved.

$$a+b=22 ab=1\times 21=21$$

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 21.

$$1,21 3,7$$

Calculate the sum for each pair.

$$1+21=22 3+7=10$$

The solution is the pair that gives sum 22.

$$a=1 b=21$$

Rewrite m^{2}+22m+21 as \left(m^{2}+m\right)+\left(21m+21\right).

$$\left(m^{2}+m\right)+\left(21m+21\right)$$

Factor out m in the first and 21 in the second group.

$$m\left(m+1\right)+21\left(m+1\right)$$

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

$$\left(m+1\right)\left(m+21\right)$$

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.

$$m^{2}+22m+21=0$$

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.

$$m=\frac{-22±\sqrt{22^{2}-4\times 21}}{2}$$

Square 22.

$$m=\frac{-22±\sqrt{484-4\times 21}}{2}$$

Multiply -4 times 21.

$$m=\frac{-22±\sqrt{484-84}}{2}$$

Add 484 to -84.

$$m=\frac{-22±\sqrt{400}}{2}$$

Take the square root of 400.

$$m=\frac{-22±20}{2}$$

Now solve the equation m=\frac{-22±20}{2} when ± is plus. Add -22 to 20.

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

Divide -2 by 2.

$$m=-1$$

Now solve the equation m=\frac{-22±20}{2} when ± is minus. Subtract 20 from -22.

$$m=-\frac{42}{2}$$

Divide -42 by 2.

$$m=-21$$

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 -21 for x_{2}.

$$m^{2}+22m+21=\left(m-\left(-1\right)\right)\left(m-\left(-21\right)\right)$$

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

$$m^{2}+22m+21=\left(m+1\right)\left(m+21\right)$$

Evaluate

$$\left(m+1\right)\left(m+21\right)$$