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Sagot :
You can use the factored form of the quadratic polynomial and then find the simplified sum.
The numerator of the simplified sum of the given expression is [tex]4x + 6[/tex]
How to factorize a quadratic polynomial with single variable?
Quadratic polynomial with single variables are expressible in the form
[tex]ax^2 + bx + c[/tex] where x is the variable and a,b,c are constants.
Its factored form is
[tex]\dfrac{1}{4a^2} \times (2ax +b-\sqrt{b^2 - 4ac})(2ax +b+ \sqrt{b^2 - 4ac})[/tex]
Using the above method and finding the simplified form of the given expression
The given expression is
[tex]\dfrac{x}{x^2 + 3x + 2} + \dfrac{3}{x + 1}[/tex]
Factorizing [tex]x^2 + 3x +2[/tex] using the aforesaid method, as we have got
a = 1, b = 3, c = 2,
thus, we have
[tex]ax^2 + bx + c = \dfrac{1}{4a^2} \times (2ax + b-\sqrt{b^2 - 4ac})(2ax +b + \sqrt{b^2 - 4ac})\\\\\begin{aligned}x^2 + 3x + 2 &= \dfrac{1}{4}(2x + 3 - \sqrt{9-8})(2x + 3 + \sqrt{9-8})\\\\&= \dfrac{1}{2}(2x+2) \times \dfrac{1}{2}(2x+4)\\&= (x+2)(x+1)\end{aligned}[/tex]
Thus, the given sum is
[tex]\begin{aligned}\dfrac{x}{x^2 + 3x + 2} + \dfrac{3}{x + 1} &= \dfrac{x}{(x+2)(x+1)} + \dfrac{3}{x + 1} \times \dfrac{(x+2)}{(x+2)}\\\\&= \dfrac{x + 3(x+2)}{(x+1)(x+2)} \\&= \dfrac{4x + 6}{(x+1)(x+2)}\\\end{aligned}[/tex]
Thus, the numerator of the simplified sum is [tex]4x + 6[/tex]
Learn more about fractions here:
https://brainly.com/question/14261303
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