`(2x^4+8x^3+7x^2-7x-12)/(x^3+4x^2+4x)`

Since the above expression is an improper rational expression , the first step is to divide and express it as a sum of simpler fractions such that the degree of polynomial in the numerator is less than the degree of polynomial in the denominator.

Dividing the above expression using...

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`(2x^4+8x^3+7x^2-7x-12)/(x^3+4x^2+4x)`

Since the above expression is an improper rational expression , the first step is to divide and express it as a sum of simpler fractions such that the degree of polynomial in the numerator is less than the degree of polynomial in the denominator.

Dividing the above expression using long division method yields,

`(2x^4+8x^3+7x^2-7x-12)/(x^3+4x^2+4x)=2x+(-x^2-7x-12)/(x^3+4x^2+4x)`

Since the polynomials do not completely divide , we have to continue with the partial fractions of the remainder expression.

Now let's factorize the denominator of the remainder expression,

`x^3+4x^2+4x=x(x^2+4x+4)`

`=x(x+2)^2`

Let, `(-x^2-7x-12)/(x^3+4x^2+4x)=A/x+B/(x+2)+C/(x+2)^2`

`=(A(x+2)^2+B(x)(x+2)+Cx)/(x(x+2)^2)`

`=(A(x^2+4x+4)+B(x^2+2x)+Cx)/(x(x+2)^2)`

`=(x^2(A+B)+x(4A+2B+C)+4A)/(x(x+2)^2)`

`:.(-x^2-7x-12)=x^2(A+B)+x(4A+2B+C)+4A`

equating the coefficients of the like terms,

`A+B=-1` ------- equation 1

`4A+2B+C=-7` ------- equation 2

`4A=-12`

`A=-12/4`

`A=-3`

Plug the value of A in equation 1,

`-3+B=-1`

`B=-1+3`

`B=2`

Plug the value of A and B in equation 2 ,

`4(-3)+2(2)+C=-7`

`-12+4+C=-7`

`C=-7+12-4`

`C=1`

`(-x^2-7x-12)/(x^3+4x^2+4x)=(-3)/x+2/(x+2)+1/(x+2)^2`

`:.(2x^4+8x^3+7x^2-7x-12)/(x^3+4x^2+4x)=2x-3/x+2/(x+2)+1/(x+2)^2`

You need to decompose the fraction in simple irreducible fractions, such that:

`(2x^4 + 8x^3 + 7x^2 - 7x - 12)/(x^3 + 4x^2 + 4x) = (2x^4 + 8x^3 + 7x^2 - 7x - 12)/(x(x^2 + 4x + 4))`

`(2x^4 + 8x^3 + 7x^2 - 7x - 12)/(x^3 + 4x^2 + 4x) = (2x^4 + 8x^3 + 7x^2 - 7x - 12)/(x(x+2)^2)`

`(2x^4 + 8x^3 + 7x^2 - 7x - 12)/(x(x+2)^2)= A/x + B/(x+2) + C/((x+2)^2)`

You need to bring to the same denominator all fractions, such that:

`2x^4 + 8x^3 + 7x^2 - 7x - 12= A(x+2)^2 + Bx(x+2) + Cx`

`2x^4 + 8x^3 + 7x^2 - 7x - 12= Ax^2 + 4Ax + 4A + Bx^2 + 2Bx + Cx`

You need to group the terms having the same power of x:

`2x^4 + 8x^3 + 7x^2 - 7x - 12= x^2(A+B) + x(4A+2B+C) + 4A`

Comparing the expressions both sides yields:

`A + B =7 `

`4A+2B+C = -7`

`4A = -12 => A = -3 => B = 7+3 => B = 10 `

`C = -7 - 4A - 2B=> C = -7 +12 - 20 => C = -15`

**Hence, the partial fraction decomposition of the improper rational expression is **`(2x^4 + 8x^3 + 7x^2 - 7x - 12)/(x(x+2)^2)= -3/x + 10/(x+2) - 15/((x+2)^2)`