# I need to factoring to simpliest form.

i am stuck on these last six problems.

2a² + a²b + ab + 1

12m²p + 35nq³

(a-2)²-1

(n²-7)(n²+1)+15

(3m+1)²-(9m²-5)

(j+6)²-(j+1)²

## Most of the expressions you have listed above are already in a simplest or factored form, or a combination of factored terms.

12m^2p + 35 nq^3 cannot be factored at all.
2a^2 + a^2*b + ab + 1
= a^2(b+2) + ab +1
OR = ab(a+1) + 2a^2 +1

## To factorize the given expressions, we will examine each one individually:

1. 2a² + a²b + ab + 1:

To factorize this expression, we can look for common factors among the terms. In this case, the common factor is 1. So, we can rewrite the expression as:

1(2a² + a²b + ab + 1)

There are no further common factors, but we notice that the first term and the last term both have a common factor of a². So, we can factor out a²:

1(a²(2 + b) + ab + 1)

Now we have factored the expression in its simplest form.

2. 12m²p + 35nq³:

Similarly, let's look for common factors among the terms. In this case, there is no common factor except for 1. So, the expression remains as:

12m²p + 35nq³

No further factoring is possible, so the expression is already in its simplest form.

3. (a - 2)² - 1:

This expression represents a difference of squares. To factorize it, we can use the formula (a - b)² = a² - 2ab + b². Applying this formula, we have:

(a - 2)² - 1 = a² - 2(a)(2) + 2² - 1

Simplifying further, we get:

a² - 4a + 4 - 1 = a² - 4a + 3

So, (a - 2)² - 1 can be factored to a² - 4a + 3.

4. (n² - 7)(n² + 1) + 15:

To factorize this expression, we can use the distributive property. Expanding the expression, we get:

n⁴ + n² - 7n² - 7 + 15

Combining like terms, we have:

n⁴ - 6n² + 8

So, (n² - 7)(n² + 1) + 15 can be factored to n⁴ - 6n² + 8.

5. (3m + 1)² - (9m² - 5):

Using the formula for the difference of squares, we can rewrite this expression as:

(3m + 1)² - (3m)² - (√5)²

Expanding the expression, we have:

9m² + 6m + 1 - 9m² - 5

Combining like terms, we get:

6m - 4

So, (3m+1)² - (9m²-5) can be factored to 6m - 4.

6. (j + 6)² - (j + 1)²:

Here, we can use the formula for the difference of squares. Expanding the expression, we have:

(j + 6)² - (j + 1)² = j² + 12j + 36 - j² - 2j - 1

Combining like terms, we get:

10j + 35

So, (j + 6)² - (j + 1)² can be factored to 10j + 35.

I hope this explanation helps you understand how to factorize these expressions and simplify them to their simplest form.