What is this expression in simplest form?

[tex] \frac{x+2}{4x^2+5x+1} \cdot \frac{4x+1}{x^2-4} [/tex]

A. [tex] \frac{1}{(x+1)(x-2)} [/tex]
B. [tex] \frac{x}{(x-2)} [/tex]
C. [tex] \frac{4x+1}{(x+1)(x-2)} [/tex]
D. [tex] \frac{4x+1}{x-2} [/tex]



Answer :

To simplify the given expression:

[tex]\[ \frac{x+2}{4 x^2+5 x+1} \cdot \frac{4 x+1}{x^2-4} \][/tex]

we should follow these steps:

1. Factor the Denominators:

First, we factor the denominators if possible.

For [tex]\(4 x^2 + 5 x + 1\)[/tex]:
We try to factor it as [tex]\((ax + b)(cx + d)\)[/tex]. We recognize it can be factored as:
[tex]\[ 4 x^2 + 5 x + 1 = (4 x + 1)(x + 1) \][/tex]

For [tex]\(x^2 - 4\)[/tex]:
Recognize it as a difference of squares:
[tex]\[ x^2 - 4 = (x - 2)(x + 2) \][/tex]

2. Rewrite the Expression:

Replace the factored forms in the original expression:
[tex]\[ \frac{x+2}{(4x+1)(x+1)} \cdot \frac{4x+1}{(x-2)(x+2)} \][/tex]

3. Simplify by Canceling Common Factors:

We notice that [tex]\((x+2)\)[/tex] and [tex]\((4x+1)\)[/tex] appear in both numerator and denominator, so they cancel out:
[tex]\[ \frac{\cancel{x+2}}{(4x+1)(x+1)} \cdot \frac{\cancel{4x+1}}{(x-2)\cancel{(x+2)}} \][/tex]

4. Simplified Form:

After canceling common factors, we get:
[tex]\[ \frac{1}{(x+1)(x-2)} \][/tex]

5. Match with Given Choices:

The simplified expression [tex]\(\frac{1}{(x+1)(x-2)}\)[/tex] matches option A.

Thus, the simplest form of the given expression is:

[tex]\[ \boxed{\frac{1}{(x+1)(x-2)}} \][/tex]