Question

use a table to multiply the polynomials. the first one has been star1. $(a - 6)(2a + 3b + 5)$step 1 - create a table and place the terms of the two polynomials along the top and left margins.step 2 - complete the table. multiply the first term in the left margin with each of the terms in the top margin and fill in the corresponding entry. repeat for each term in the left margin.step 3 - write up an expression for the sum of all the products and simplify the expression.2. $(7x + 2)(x^2 + 5x + 1)$step 1 - create a table and place the terms of the two polynomials along the top and left margins.step 2 - create and complete the table. multiply the first term in the left margin with each of the terms in the top margin and fill in the corresponding entry. repeat for each term in the left margin.step 3 - write up an expression for the sum of all the products and simplify the expression.

Explanation:

Step 1: Solve $(a-6)(2a+3b+5)$

Step 1a: Fill table with products

Multiply left terms by top terms:

	$+2a$	$+3b$	$+5$
$-6$	$-6 \cdot 2a = -12a$	$-6 \cdot 3b = -18b$	$-6 \cdot 5 = -30$

Step 1b: Sum and simplify terms

Combine all products and like terms:
$2a^2 + 3ab + 5a - 12a - 18b - 30 = 2a^2 + 3ab -7a -18b -30$

Step 2: Solve $(7x+2)(x^2+5x+1)$

Step 2a: Fill table with products

Multiply left terms by top terms:

	$x^2$	$5x$	$1$
$2$	$2 \cdot x^2 = 2x^2$	$2 \cdot 5x = 10x$	$2 \cdot 1 = 2$

Step 2b: Sum and simplify terms

Combine all products and like terms:
$7x^3 + 35x^2 + 7x + 2x^2 + 10x + 2 = 7x^3 + 37x^2 +17x +2$

Answer:

1. $2a^2 + 3ab -7a -18b -30$
2. $7x^3 + 37x^2 +17x +2$