Solve 4x^2+5x+1

$$4 { x }^{ 2 } +5x+1$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=5 ab=4\times 1=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=1 b=4$$

Rewrite 4x^{2}+5x+1 as \left(4x^{2}+x\right)+\left(4x+1\right).

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

Factor out x in 4x^{2}+x.

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

Factor out common term 4x+1 by using distributive property.

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

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.

$$4x^{2}+5x+1=0$$

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.

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

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 4}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-5±\sqrt{25-16}}{2\times 4}$$

Add 25 to -16.

$$x=\frac{-5±\sqrt{9}}{2\times 4}$$

Take the square root of 9.

$$x=\frac{-5±3}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-5±3}{8}$$

Now solve the equation x=\frac{-5±3}{8} when ± is plus. Add -5 to 3.

$$x=-\frac{2}{8}$$

Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{4}$$

Now solve the equation x=\frac{-5±3}{8} when ± is minus. Subtract 3 from -5.

$$x=-\frac{8}{8}$$

Divide -8 by 8.

$$x=-1$$

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

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

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

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

Add \frac{1}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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

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

Evaluate

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