What is the simplified form of 3/(4x + 3) + 21/(8x² + 26x + 15)

Solution:

Step 1:

First we need to factorise 8x² + 26x + 15 so that LCM is possible to take for both the terms.

On factorising the term 8x² + 26x + 15 by factorization method,

The factors of product 120(8 × 15) are:

20 and 6

So that 20 + 6 gives a middle term that is 26,

which is the rule of factorization of quadratic equation

Step2:

⇒ 3/(4x + 3) + 21 / (8x² + 26x + 15)

= 3/(4x + 3) + 21/(2x + 5)(4x + 3)

In the first term only 4x + 3 is the term in the denominator which is need to multiplied by (2x + 5) with both numerator and denominator

⇒3/(4x + 3) + 21/(2x + 5)(4x + 3)

= [3(2x + 5)/(4x + 3)(2x + 5)] + [21/(2x + 5)(4x + 3)]

Step3:

After taking LCM for both the terms

⇒ 3/(4x + 3) + 21/(2x + 5)(4x + 3) = [3(2x + 5) + 21 ] / (4x + 3)(2x + 5)

⇒ 3/(4x + 3) + 21/(2x + 5)(4x + 3) = [6x + 15 + 21 ]/(4x + 3) (2x + 5)

⇒ 3/(4x + 3) + 21/(2x + 5)(4x + 3) = [6x + 36 ]/(4x + 3)(2x + 5)

⇒ 3/(4x + 3) + 21/(2x + 5)(4x + 3) = 6(x + 6)/(4x + 3)(2x + 5)

Therefore, the simplified form is 6(x + 6)/(4x + 3)(2x + 5)