Simplify the following expression completely. (x^(2)+8x+15)/(x^(2)+7x+12) Answer:

Factor Numerator: Factor the numerator x^2 + 8x + 15. To factor a quadratic expression, we look for two numbers that multiply to the constant term (15) and add up to the coefficient of the x term (8). The numbers that satisfy these conditions are 3 and 5. So, x^2 + 8x + 15 can be factored as (x + 3)(x + 5).Factor Denominator: Factor the denominator x^2 + 7x + 12. Similarly, we look for two numbers that multiply to the constant term (12) and add up to the coefficient of the x term (7). The numbers that satisfy these conditions are 3 and 4. So, x^2 + 7x + 12 can be factored as (x + 3)(x + 4).Write Factored Form: Write the factored form of the numerator and denominator. The expression becomes: ((x + 3)(x + 5))/((x + 3)(x + 4))Cancel Common Factors: Cancel out the common factors. The (x + 3) term is present in both the numerator and the denominator, so we can cancel it out. The simplified expression is: (x + 5)/(x + 4)

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Q. Simplify the following expression completely.

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\frac{x^{2}+8 x+15}{x^{2}+7 x+12}

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Answer:

Factor Numerator: Factor the numerator $x^2 + 8x + 15$ . To factor a quadratic expression, we look for two numbers that multiply to the constant term $(15)$ and add up to the coefficient of the $x$ term $(8)$ . The numbers that satisfy these conditions are $3$ and $5$ . So, $x^2 + 8x + 15$ can be factored as $(x + 3)(x + 5)$ .
Factor Denominator: Factor the denominator $x^2 + 7x + 12$ . Similarly, we look for two numbers that multiply to the constant term $(12)$ and add up to the coefficient of the $x$ term $(7)$ . The numbers that satisfy these conditions are $3$ and $4$ . So, $x^2 + 7x + 12$ can be factored as $(x + 3)(x + 4)$ .
Write Factored Form: Write the factored form of the numerator and denominator. $\newline$ The expression becomes: $\newline$ $\frac{(x + 3)(x + 5)}{(x + 3)(x + 4)}$
Cancel Common Factors: Cancel out the common factors. $\newline$ The $(x + 3)$ term is present in both the numerator and the denominator, so we can cancel it out. $\newline$ The simplified expression is: $\newline$ $\frac{x + 5}{x + 4}$