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(a^(2)-b^(2)+2a+1)/(a+1+b)= ?
A)
a+3b+1
B)
3a-b+1
C)
a-b+1
D)
a-b+3
E)
a+b-1

$40$ . $\frac{a^{2}-b^{2}+2 a+1}{a+1+b}=$ ? $\newline$ A) $a+3 b+1$ $\newline$ B) $3 a-b+1$ $\newline$ C) $a-b+1$ $\newline$ D) $a-b+3$ $\newline$ E) $a+b-1$

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Factor sums and differences of cubes

Full solution

Q.

40

.

\frac{a^{2}-b^{2}+2 a+1}{a+1+b}=

?

\newline

A)

a+3 b+1

\newline

B)

3 a-b+1

\newline

C)

a-b+1

\newline

D)

a-b+3

\newline

E)

a+b-1

Recognize Quadratic Expression: First, recognize that the numerator is a quadratic expression that can be factored. The term $a^2 - b^2$ is a difference of squares, which factors into $(a - b)(a + b)$ . The remaining terms $2a + 1$ do not factor with the difference of squares, so we will keep them separate for now.
Factor Numerator: Now, let's factor the numerator: $a^2 - b^2 + 2a + 1 = (a - b)(a + b) + 2a + 1$ .
Look for Common Factors: Next, we need to look for common factors between the numerator and the denominator. The denominator is $a + 1 + b$ , which can be reordered as $(a + b + 1)$ to make it easier to compare with the numerator.
Expand and Simplify Numerator: We can see that there is no common factor between $(a - b)(a + b) + 2a + 1$ and $a + b + 1$ , so we cannot simplify the expression by canceling out any terms. Therefore, we need to expand the numerator and then simplify the expression.
Factor Perfect Square Trinomial: Let's expand the numerator: $(a - b)(a + b) + 2a + 1 = a^2 - b^2 + 2a + 1$ .
Rewrite with Factored Numerator: Now, we will simplify the numerator by combining like terms: $a^2 - b^2 + 2a + 1 = a^2 + 2a + 1 - b^2$ .
Simplify by Canceling Common Factor: Notice that $a^2 + 2a + 1$ is a perfect square trinomial, which factors into $(a + 1)^2$ . So the numerator becomes $(a + 1)^2 - b^2$ .
Final Simplified Expression: We can now rewrite the expression with the factored numerator: $((a + 1)^2 - b^2) / (a + b + 1)$ .
Final Simplified Expression: We can now rewrite the expression with the factored numerator: $\frac{(a + 1)^2 - b^2}{a + b + 1}$ .The numerator is now a difference of squares again, with $(a + 1)^2$ being the square of $(a + 1)$ and $b^2$ being the square of $b$ . We can factor the numerator as $((a + 1) - b)((a + 1) + b)$ .
Final Simplified Expression: We can now rewrite the expression with the factored numerator: $\frac{(a + 1)^2 - b^2}{a + b + 1}$ .The numerator is now a difference of squares again, with $(a + 1)^2$ being the square of $(a + 1)$ and $b^2$ being the square of $b$ . We can factor the numerator as $((a + 1) - b)((a + 1) + b)$ .After factoring, the expression becomes: $\frac{(a + 1 - b)(a + 1 + b)}{a + b + 1}$ .
Final Simplified Expression: We can now rewrite the expression with the factored numerator: $((a + 1)^2 - b^2) / (a + b + 1)$ .The numerator is now a difference of squares again, with $(a + 1)^2$ being the square of $(a + 1)$ and $b^2$ being the square of $b$ . We can factor the numerator as $((a + 1) - b)((a + 1) + b)$ .After factoring, the expression becomes: $((a + 1 - b)(a + 1 + b)) / (a + b + 1)$ .Now, we can simplify the expression by canceling out the common factor of $(a + b + 1)$ in the numerator and denominator. However, we must be careful here because the numerator has $(a + 1 - b)$ and $(a + 1 + b)$ , which are not the same as $(a + b + 1)$ . There is no common factor to cancel out, so the expression cannot be simplified further.
Final Simplified Expression: We can now rewrite the expression with the factored numerator: $((a + 1)^2 - b^2) / (a + b + 1)$ . The numerator is now a difference of squares again, with $(a + 1)^2$ being the square of $(a + 1)$ and $b^2$ being the square of $b$ . We can factor the numerator as $((a + 1) - b)((a + 1) + b)$ . After factoring, the expression becomes: $((a + 1 - b)(a + 1 + b)) / (a + b + 1)$ . Now, we can simplify the expression by canceling out the common factor of $(a + b + 1)$ in the numerator and denominator. However, we must be careful here because the numerator has $(a + 1 - b)$ and $(a + 1 + b)$ , which are not the same as $(a + b + 1)$ . There is no common factor to cancel out, so the expression cannot be simplified further. The final simplified expression is $((a + 1 - b)(a + 1 + b)) / (a + b + 1)$ , which does not match any of the answer choices provided. It seems there might have been a mistake in the simplification process. Let's re-evaluate the expression.

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