Question

sari is factoring the polynomial $2x^2 + 5x + 3$. if one factor is $(x + 1)$, what is the other factor?
\bigcirc\\ $2x - 3$
\bigcirc\\ $2x + 3$
\bigcirc\\ $3x - 2$
\bigcirc\\ $3x + 2$

Explanation:

Step1: Recall polynomial factoring

To find the other factor, we can perform polynomial division or use the fact that if \((x + 1)\) is a factor of \(2x^2+5x + 3\), then we can factor the quadratic as \((x + 1)(ax + b)\) and find \(a\) and \(b\). Expanding \((x + 1)(ax + b)\) gives \(ax^2+(a + b)x + b\). Comparing with \(2x^2+5x + 3\), we have \(a = 2\) (from the \(x^2\) term) and \(b = 3\) (from the constant term, since \(1\times b=3\)). Then check the middle term: \(a + b=2 + 3 = 5\), which matches the middle term of \(2x^2+5x + 3\). So the other factor is \(2x + 3\).

Step2: Verify by multiplication

Multiply \((x + 1)\) and \((2x + 3)\):

$$
LATEXBLOCK0
$$

This matches the original polynomial, so the other factor is correct.

Answer:

B. \(2x + 3\)