Solved on Feb 02, 2024

Expand the expression $(a+2b)(4a+b)$ .

STEP 1

Assumptions
1. We are given a binomial expression $(a+2b)$ .
2. We are given another binomial expression $(4a+b)$ .
3. We need to multiply these two binomials together.

STEP 2

To multiply two binomials, we use the distributive property (also known as the FOIL method in the context of binomials), which states that for any numbers or expressions $x$ , $y$ , $z$ , and $w$ :
$(x + y)(z + w) = xz + xw + yz + yw$

STEP 3

Using the distributive property, we multiply each term in the first binomial by each term in the second binomial.
$(a+2b)(4a+b) = a(4a) + a(b) + 2b(4a) + 2b(b)$

STEP 4

Calculate each product.
$a(4a) = 4a^2$ $a(b) = ab$ $2b(4a) = 8ab$ $2b(b) = 2b^2$

STEP 5

Combine the products to get the final expanded form.
$4a^2 + ab + 8ab + 2b^2$

STEP 6

Combine like terms. In this case, $ab$ and $8ab$ are like terms.
$4a^2 + (1 + 8)ab + 2b^2$

STEP 7

Add the coefficients of the like terms.
$4a^2 + 9ab + 2b^2$

STEP 8

Write the final simplified expression.
The product of the two binomials $(a+2b)(4a+b)$ is $4a^2 + 9ab + 2b^2$ .

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Expand the expression $(a+2b)(4a+b)$ .

STEP 1

Assumptions
1. We are given a binomial expression $(a+2b)$ .
2. We are given another binomial expression $(4a+b)$ .
3. We need to multiply these two binomials together.

STEP 2

To multiply two binomials, we use the distributive property (also known as the FOIL method in the context of binomials), which states that for any numbers or expressions $x$ , $y$ , $z$ , and $w$ :
$(x + y)(z + w) = xz + xw + yz + yw$

STEP 3

Using the distributive property, we multiply each term in the first binomial by each term in the second binomial.
$(a+2b)(4a+b) = a(4a) + a(b) + 2b(4a) + 2b(b)$

STEP 4

Calculate each product.
$a(4a) = 4a^2$ $a(b) = ab$ $2b(4a) = 8ab$ $2b(b) = 2b^2$

STEP 5

Combine the products to get the final expanded form.
$4a^2 + ab + 8ab + 2b^2$

STEP 6

Combine like terms. In this case, $ab$ and $8ab$ are like terms.
$4a^2 + (1 + 8)ab + 2b^2$

STEP 7

Add the coefficients of the like terms.
$4a^2 + 9ab + 2b^2$

STEP 8

Write the final simplified expression.
The product of the two binomials $(a+2b)(4a+b)$ is $4a^2 + 9ab + 2b^2$ .

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Expand the expression (a+2b)(4a+b)(a+2b)(4a+b)(a+2b)(4a+b).

STEP 1

Assumptions1. We are given a binomial expression (a+2b)(a+2b)(a+2b).2. We are given another binomial expression (4a+b)(4a+b)(4a+b).3. We need to multiply these two binomials together.

STEP 2

To multiply two binomials, we use the distributive property (also known as the FOIL method in the context of binomials), which states that for any numbers or expressions xxx, yyy, zzz, and www:(x+y)(z+w)=xz+xw+yz+yw(x + y)(z + w) = xz + xw + yz + yw(x+y)(z+w)=xz+xw+yz+yw

STEP 3

Using the distributive property, we multiply each term in the first binomial by each term in the second binomial.(a+2b)(4a+b)=a(4a)+a(b)+2b(4a)+2b(b)(a+2b)(4a+b) = a(4a) + a(b) + 2b(4a) + 2b(b)(a+2b)(4a+b)=a(4a)+a(b)+2b(4a)+2b(b)

STEP 4

Calculate each product.a(4a)=4a2a(4a) = 4a^2a(4a)=4a2 a(b)=aba(b) = aba(b)=ab 2b(4a)=8ab2b(4a) = 8ab2b(4a)=8ab 2b(b)=2b22b(b) = 2b^22b(b)=2b2

STEP 5

Combine the products to get the final expanded form.4a2+ab+8ab+2b24a^2 + ab + 8ab + 2b^24a2+ab+8ab+2b2

STEP 6

Combine like terms. In this case, ababab and 8ab8ab8ab are like terms.4a2+(1+8)ab+2b24a^2 + (1 + 8)ab + 2b^24a2+(1+8)ab+2b2

STEP 7

Add the coefficients of the like terms.4a2+9ab+2b24a^2 + 9ab + 2b^24a2+9ab+2b2

STEP 8

Write the final simplified expression.The product of the two binomials (a+2b)(4a+b)(a+2b)(4a+b)(a+2b)(4a+b) is 4a2+9ab+2b24a^2 + 9ab + 2b^24a2+9ab+2b2.

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Expand the expression $(a+2b)(4a+b)$ .

Assumptions
1. We are given a binomial expression $(a+2b)$ .
2. We are given another binomial expression $(4a+b)$ .
3. We need to multiply these two binomials together.

To multiply two binomials, we use the distributive property (also known as the FOIL method in the context of binomials), which states that for any numbers or expressions $x$ , $y$ , $z$ , and $w$ :
$(x + y)(z + w) = xz + xw + yz + yw$

Using the distributive property, we multiply each term in the first binomial by each term in the second binomial.
$(a+2b)(4a+b) = a(4a) + a(b) + 2b(4a) + 2b(b)$

Calculate each product.
$a(4a) = 4a^2$ $a(b) = ab$ $2b(4a) = 8ab$ $2b(b) = 2b^2$

Combine the products to get the final expanded form.
$4a^2 + ab + 8ab + 2b^2$

Combine like terms. In this case, $ab$ and $8ab$ are like terms.
$4a^2 + (1 + 8)ab + 2b^2$

Add the coefficients of the like terms.
$4a^2 + 9ab + 2b^2$

Write the final simplified expression.
The product of the two binomials $(a+2b)(4a+b)$ is $4a^2 + 9ab + 2b^2$ .