### Theory:

Follow the steps to multiply the polynomials in the horizontal method:
• First, write the $$2$$ polynomials in a row separated by using a multiplication sign.
• Multiply each term of the first polynomial with each term of the second polynomial.
• In this, the product is obtained by combining and adding the like terms.
Example:
We learn how to multiply two binomials $$a + 5$$ by $$a + 7$$ using a horizontal method.

Consider the polynomials $$a + 5$$ and $$a + 7$$.

Separate the two polynomials using a multiplication sign.

$= \left(a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5\right)\phantom{\rule{0.147em}{0ex}}×\phantom{\rule{0.147em}{0ex}}\left(a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}7\right)$

Multiply each term of the first polynomial with each term of the second polynomial.

$=\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}×\phantom{\rule{0.147em}{0ex}}\left(a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}7\right) +\phantom{\rule{0.147em}{0ex}}5×\phantom{\rule{0.147em}{0ex}}\left(a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}7\right)$

$=\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}×a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}×\phantom{\rule{0.147em}{0ex}}7\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5\phantom{\rule{0.147em}{0ex}}×a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5\phantom{\rule{0.147em}{0ex}}×\phantom{\rule{0.147em}{0ex}}7$

Combine the like term.

$=\phantom{\rule{0.147em}{0ex}}{a}^{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}7a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}5a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}35$

$=\phantom{\rule{0.147em}{0ex}}{a}^{2}+\phantom{\rule{0.147em}{0ex}}12a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}35$

Therefore, the multiplication of the given polynomials is ${a}^{2}+\phantom{\rule{0.147em}{0ex}}12a\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}35$.