With Coefficients

Coefficients and signs change the arithmetic, but the method stays the same. To expand $\left(2x+1\right)\left(x-3\right)$, write the four products with their signs. The first two are $2xx = 2x^{2}$ and $2x\cdot \left(-3\right) = -6x$. The last two are $x = x$ and $-3 = -3$.

Collect the $x$ terms with their signs: $-6x+x = -5x$. The product is $2x^{2}-5x-3$.

When the $x$ terms have coefficients other than 1, the middle coefficient depends on both cross products. Multiply each term of one binomial by each term of the other. Then add the two $x$ terms, keeping their signs.