Simplify completely quantity x squared plus 4 x minus 45 all over x squared plus 10 x plus 9 and find the restrictions on the variable.

, x ≠ −1, x ≠ −9

Step-by-step explanation:

To simplify this, we must first factor both the numerator and the denominator:

They can be factored by using grouping.

The numerator expands to:

x^2 + 9x - 5x - 45

(x^2+9x) + (-5x-45)

x(x+9) - 5(x+9)

(x-5)(x+9)

The denominator expands to:

x^2 + 9x + x + 9

(x^2 + 9x) + (x + 9)

x(x+9) + 1(x+9)

(x+1)(x+9)

So now it looks like this:

Now that it is fully factored (albeit not simplified), we can determine which values of x would be restricted from this function.

Since the factors of the denominator are x + 1 and x + 9, we know that values of x that would make them 0 are going to be restrictions.

(x + 1)(x + 9) = 0

x + 1 = 0; x = -1

x + 9 = 0; x = -9

Because these values of x make the denominator undefined, they are restrictions. An undefined denominator makes the entire answer undefined, which proves to be a problem, right?

And lastly, to simplify the answer, we can divide (x + 9) by (x + 9) and they will cancel out because they are the same. That leaves the answer to be:

x - 5 / x + 1

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Have a nice day, you all. Remember, you're not alone - always someone there for you.

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step-by-step explanation: