Think of the minus sign before $(5x^2 - 7x - 1)$ as multiplying by $-1$. How does that affect each term inside the parentheses?

After you have expanded both parts, collect the $x^2$ terms, the $x$ terms, and the constant terms separately. What is the combined coefficient on $x^2$?

You only need the coefficient of $x^2$. You can ignore the exact values of the $x$ term and the constant once you know the $x^2$ coefficient.

In Desmos, type the expression as a function, for example: f(x) = 2(3x^2 - x + 4) - (5x^2 - 7x - 1) and look at the graph.

After you simplify the expression by hand into a trinomial $px^2 + qx + r$, enter that as a second function, for example g(x) = px^2 + qx + r using your values of $p$, $q$, and $r$.

Use a table (tap the gear icon and select "Table") or just look at the graphs for several $x$-values (such as $x=-1, 0, 2$). If the $y$-values of $f(x)$ and $g(x)$ match for all tested $x$, then your simplification—and thus your value of $p$—is consistent. If they differ, re-check your distribution and combination of like terms.

Multiply each term inside the first parentheses by 2:

A minus sign in front of parentheses is the same as multiplying by $-1$. Change the sign of every term inside the second parentheses:

Now add the two results together:

Group like terms (same power of $x$):

• $x^2$ terms: $6x^2 - 5x^2 = x^2$
• $x$ terms: $-2x + 7x = 5x$
• constant terms: $8 + 1 = 9$

So the expression becomes

The simplified expression is $x^2 + 5x + 9$, which matches $px^2 + qx + r$.

Here, the coefficient of $x^2$ is $1$, so $p = 1$.