For a linear factor like $2x+5$ or $x+5$, first solve it for $x$ (for example, $2x+5=0$ gives $x=-\tfrac{5}{2}$). Then see what the original polynomial equals at that $x$-value.

Write $P(x)=2x^3-5x^2-18x+45$ and plug in the $x$-values you found from the linear answer choices. Which substitution makes $P(x)$ equal zero?

Try grouping the terms of the polynomial as $(2x^3-5x^2)+(-18x+45)$ and factor out the greatest common factor from each group. You should see the same binomial appear in both groups, which will be a factor.

In Desmos, type f(x) = 2x^3 - 5x^2 - 18x + 45 to define the polynomial.

In new lines, type f(-5/2), f(5/2), and f(-5) (these come from $2x+5$, $2x-5$, and $x+5$ respectively). Look at the outputs: the $x$-value that makes $f(x)$ equal 0 corresponds to the linear factor from the choices that matches $0$ at that $x$-value.

A linear expression like $ax+b$ is a factor of a polynomial $P(x)$ if and only if $P\big(-\tfrac{b}{a}\big)=0$.

So for each linear answer choice, we can:

• Set it equal to $0$ to find its corresponding $x$-value (its root).
• Plug that $x$-value into the polynomial.
• If the result is $0$, that expression is a factor.

Look at each linear answer choice and solve for $x$:

• If $2x+5=0$, then $x=-\tfrac{5}{2}$.
• If $2x-5=0$, then $x=\tfrac{5}{2}$.
• If $x+5=0$, then $x=-5$.

(The choice $x^2+9$ is quadratic, not linear, so it doesn’t correspond to a single specific $x$-value; we will first test the easier linear options.)

Let the polynomial be

We will evaluate $P(x)$ at each of the candidate roots from Step 2:

• $P\big(-\tfrac{5}{2}\big)$
• $P\big(\tfrac{5}{2}\big)$
• $P(-5)$

Whichever $x$-value makes $P(x)=0$ will tell us which linear expression is a factor.

Compute each value carefully:

For $x=-\tfrac{5}{2}$ (from $2x+5$):

So $2x+5$ is not a factor.

For $x=-5$ (from $x+5$):

So $x+5$ is not a factor.

For $x=\tfrac{5}{2}$ (from $2x-5$):

Since $P\big(\tfrac{5}{2}\big)=0$, the Factor Theorem tells us that the corresponding linear expression $2x-5$ is a factor of the polynomial. So the correct answer is $2x-5$.