Question

Completely factor the expression below. 4x^2 + 14x + 10

Answer 1

`2(x+1)(2x+5)`

Explanation:

First, factor out 2:

`2(2x^2+7x+5)`

Consider
`2(2x^2+7x+5)`. Factor the expression by grouping. First, the expression needs to be rewritten as
`2x^2+ax+bx+5`. To find a and b, set up a system to be solved:

`a + b = 7`

`ab = 2 * 5 = 10`

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 10:

`1, 10`

`2, 5`

Calculate the sum for each pair:

`1 + 10 = 11`

`2 + 5 = 7`

The solution is the pair that gives sum 7:

`a = 2`

`b = 5`

Rewrite
`2x^2+7x+5` as
`(2x^2+2x)+(5x+5):`

`(2x^2+2x)+(5x+5)`

Factor out 2x in the first and 5 in the second group:

`2x(x+1)+5(x+1)`

Factor out common term x + 1 by using distributive property:

`(x+1)(2x+5)`

Rewrite the complete factored expression:

`2(x+1)(2x+5)`

Answer 2

Answer:

We can start by factoring out a common factor of 2 from all three terms:

4x^2 + 14x + 10 = 2(2x^2 + 7x + 5)

Now we need to factor the quadratic expression inside the parentheses. We can use the fact that we want two numbers that multiply to 5*2=10 and add to 7.

The two numbers are 2 and 5. We can use these two numbers to rewrite the quadratic expression as follows:


2x^2 + 7x + 5 = 2x^2 + 2x + 5x + 5
= 2x(x+1) + 5(x+1)
= (2x+5)(x+1)

Therefore, the fully factored form of the expression is:

4x^2 + 14x + 10 = 2(2x+5)(x+1)