Matching coefficients of a polynomial equation

If an equation holds for every value of $x$, the two sides must be the exact same expression.
That can only happen when each power of $x$ has the same coefficient on both sides.
So you set corresponding coefficients equal and solve for the unknown constants.

This means you never plug in a value for $x$.
Instead, you line up the $x^2$ terms, the $x$ terms, and the constants separately.
Each match gives you one equation to work with.

Take $-18x - 24y^2 = 3mx + 4my^2$, true for all $x$ and $y$.
Match the $x$ terms: $-18 = 3m$, so $m = -6$.
Match the $y^2$ terms: $-24 = 4m$, which also gives $m = -6$, confirming the answer.

If a power is missing on one side, its coefficient there is $0$.
In $4x^3 - 5x - 9$ there is no $x^2$ term, so its coefficient is $0$.
Matching that against $(m - 2)x^2$ gives $m - 2 = 0$, so $m = 2$.

If $ax + b = 7x - 4$ for all $x$, find $a$ and $b$.
Match the $x$ coefficients: $a = 7$.
Match the constants: $b = -4$.

Comparing $x^2 + y^2 - 8x + 10y - 8 = 0$ with $x^2 + y^2 - 2ax - 2by - 8 = 0$, find $a$ and $b$.
Match the $x$ terms: $-2a = -8$, so $a = 4$.
Match the $y$ terms: $-2b = 10$, so $b = -5$.

If $(x + 3)(x + k) = x^2 + 7x + 12$, find $k$.
Expand the left side: $x^2 + (3 + k)x + 3k$.
Match the $x$ terms: $3 + k = 7$, so $k = 4$.