Solve (5x^2+21x+4)

$$( 5 x ^ { 2 } + 21 x + 4 )$$

Factor

$$\left(x+4\right)\left(5x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=21 ab=5\times 4=20$$

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 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 21.

$$a=1 b=20$$

Rewrite 5x^{2}+21x+4 as \left(5x^{2}+x\right)+\left(20x+4\right).

$$\left(5x^{2}+x\right)+\left(20x+4\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(5x+1\right)+4\left(5x+1\right)$$

Factor out common term 5x+1 by using distributive property.

$$\left(5x+1\right)\left(x+4\right)$$

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.

$$5x^{2}+21x+4=0$$

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.

$$x=\frac{-21±\sqrt{21^{2}-4\times 5\times 4}}{2\times 5}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\times 5\times 4}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-21±\sqrt{441-20\times 4}}{2\times 5}$$

Multiply -20 times 4.

$$x=\frac{-21±\sqrt{441-80}}{2\times 5}$$

Add 441 to -80.

$$x=\frac{-21±\sqrt{361}}{2\times 5}$$

Take the square root of 361.

$$x=\frac{-21±19}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-21±19}{10}$$

Now solve the equation x=\frac{-21±19}{10} when ± is plus. Add -21 to 19.

$$x=-\frac{2}{10}$$

Reduce the fraction \frac{-2}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{5}$$

Now solve the equation x=\frac{-21±19}{10} when ± is minus. Subtract 19 from -21.

$$x=-\frac{40}{10}$$

Divide -40 by 10.

$$x=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{5} for x_{1} and -4 for x_{2}.

$$5x^{2}+21x+4=5\left(x-\left(-\frac{1}{5}\right)\right)\left(x-\left(-4\right)\right)$$

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

$$5x^{2}+21x+4=5\left(x+\frac{1}{5}\right)\left(x+4\right)$$

Add \frac{1}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5x^{2}+21x+4=5\times \frac{5x+1}{5}\left(x+4\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5x^{2}+21x+4=\left(5x+1\right)\left(x+4\right)$$

Evaluate

$$\left(x+4\right)\left(5x+1\right)$$