Treat $a^2$, $ab$, and $2b$ as three separate pieces. Expand $(2t - 3)^2$, expand $(2t - 3)(5 - t)$, and distribute in $2(5 - t)$, watching the signs.

In $a^2 - ab + 2b$, the minus applies to the entire product $ab$. After you expand $ab$, are you changing the sign of every term when you subtract it?

Once everything is expanded and the subtraction of $ab$ is handled, collect $t^2$ terms together, $t$ terms together, and constants together. Then look for the answer choice with exactly those coefficients.

In Desmos, replace $t$ with $x$ and type the original expression as

$(2x - 3)^2 - (2x - 3)(5 - x) + 2(5 - x)$

Label it as something like $f(x)$ so you know it is the original expression.

On new lines, enter each option as a function of $x$ (for example, $g(x) = 6x^2 - 25x + 34$, $h(x) = 6x^2 - 27x + 24$, and so on). Create a table for $f(x)$ and plug in a few easy $x$ values such as 0, 1, and 2. For each of those $x$ values, see which option’s function gives the same $y$ values as $f(x)$ every time; that option is equivalent to the original expression.

We are given $a = 2t - 3$ and $b = 5 - t$ and asked to simplify $a^2 - ab + 2b$.

Substitute the expressions for $a$ and $b$:

Now we will simplify this step by step.

First expand $a^2$:

Next expand $ab$ using distribution (FOIL):

Then expand $2b$:

Now replace $a^2$, $ab$, and $2b$ in the original expression with their expanded forms:

Handle the minus sign in front of $ab$ carefully. Subtracting $ab$ means you must change the sign of each term in $ab$:

So the whole expression becomes

Now combine like terms.

Group like terms:

• $t^2$ terms: $4t^2 + 2t^2 = 6t^2$
• $t$ terms: $-12t - 13t - 2t = -27t$
• Constant terms: $9 + 15 + 10 = 34$

So the simplified expression is

This matches choice C.