Question

rational expressions
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simplify: \\(\frac{3x^2 + 11x + 10}{3x^2 + 20x + 25}\\)
\\(\frac{2}{5}\\)
\\(\frac{11x + 10}{20x + 25}\\)
\\(\frac{3x + 2}{3x + 5}\\)
\\(\frac{x + 2}{x + 5}\\)
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Explanation:

Step1: Factor numerator

$3x^2 + 11x + 10 = (3x + 5)(x + 2)$

Step2: Factor denominator

$3x^2 + 20x + 25 = (3x + 5)(3x + 5)$

Step3: Cancel common factors

$\frac{(3x + 5)(x + 2)}{(3x + 5)(3x + 5)} = \frac{x + 2}{3x + 5}$ Wait correction: Wait numerator factoring mistake—correct numerator: $3x^2+11x+10$: find two numbers multiplying to $3×10=30$ and adding to 11: 5&6. Split: $3x²+6x+5x+10=3x(x+2)+5(x+2)=(3x+5)(x+2)$? No wait denominator is $(3x+5)^2$. Wait wait another way: Wait no, wait the options have $\frac{3x+2}{3x+5}$—wait did I factor wrong? Wait $3x²+11x+10$: let's check $(3x+2)(x+5)=3x²+15x+2x+10=3x²+17x+10$ no. $(3x+5)(x+2)=3x²+6x+5x+10=3x²+11x+10$ yes. Denominator $3x²+20x+25=(3x+5)(x+5)$? Wait $3x×x=3x²$, $3x×5+5×x=15x+5x=20x$, $5×5=25$ yes! Oh my mistake earlier denominator was $(3x+5)(x+5)$ not squared. So now cancel $(3x+5)$: $\frac{x+2}{x+5}$? Wait no wait option D is $\frac{x+2}{x+5}$. Wait wait let's recheck:

Numerator: $3x²+11x+10$: discriminant $121-120=1$, roots $(-11±1)/6$ → $(-10)/6=-5/3$, $(-12)/6=-2$. So factors: $3(x + 5/3)(x + 2)=(3x+5)(x+2)$ correct.

Denominator: $3x²+20x+25$: discriminant $400-300=100$, roots $(-20±10)/6$ → $(-10)/6=-5/3$, $(-30)/6=-5$. So factors: $3(x +5/3)(x+5)=(3x+5)(x+5)$ correct.

Cancel $(3x+5)$: $\frac{x+2}{x+5}$ which is option D. Wait earlier mistake in denominator factoring (thought it was squared but no). So correct answer is D.

Step1: Factor numerator

$3x^2 +11x+10=(3x+5)(x+2)$

Step2: Factor denominator

$3x^2 +20x+25=(3x+5)(x+5)$

Step3: Cancel common factor

$\frac{(3x+5)(x+2)}{(3x+5)(x+5)}=\frac{x+2}{x+5}$

Answer:

C. $\frac{3x + 2}{3x + 5}$