Question

1. $x^2 + xy + 8x + 8y$ 6. $q^2 + q + 5pq + 5p$ 7. $m^2 - 3n - 3m + mn$ 8. $2a^2 - 12b + 8ab - 3a$

Explanation:

Let's solve each of these polynomial factorization problems one by one. We'll use the method of grouping to factor each polynomial.

Problem 5: \( x^2 + xy + 8x + 8y \)

Step 1: Group the terms

Group the first two terms and the last two terms:
\( (x^2 + xy) + (8x + 8y) \)

Step 2: Factor out the GCF from each group

From the first group, factor out \( x \): \( x(x + y) \)
From the second group, factor out \( 8 \): \( 8(x + y) \)
So now we have: \( x(x + y) + 8(x + y) \)

Step 3: Factor out the common binomial factor

The common binomial factor is \( (x + y) \), so we factor that out:
\( (x + y)(x + 8) \)

Step 1: Group the terms

Group the first two terms and the last two terms:
\( (q^2 + q) + (5pq + 5p) \)

Step 2: Factor out the GCF from each group

From the first group, factor out \( q \): \( q(q + 1) \)
From the second group, factor out \( 5p \): \( 5p(q + 1) \)
So now we have: \( q(q + 1) + 5p(q + 1) \)

Step 3: Factor out the common binomial factor

The common binomial factor is \( (q + 1) \), so we factor that out:
\( (q + 1)(q + 5p) \)

Step 1: Rearrange and group the terms

First, rearrange the terms to group like terms:
\( (m^2 - 3m) + (mn - 3n) \)

Step 2: Factor out the GCF from each group

From the first group, factor out \( m \): \( m(m - 3) \)
From the second group, factor out \( n \): \( n(m - 3) \)
So now we have: \( m(m - 3) + n(m - 3) \)

Step 3: Factor out the common binomial factor

The common binomial factor is \( (m - 3) \), so we factor that out:
\( (m - 3)(m + n) \)

Answer:

\( (x + y)(x + 8) \)

Problem 6: \( q^2 + q + 5pq + 5p \)