Question

1. $6x^2 + 26x + 8$

8.

1. $3x^2 + 9xy + 6y^2$

11.

1. $2x^2 + 15xy + 25y^2$

14.

Explanation:

Problem 7: \(6x^2 + 26x + 8\)

Step 1: Factor out the GCF

The greatest common factor (GCF) of \(6x^2\), \(26x\), and \(8\) is \(2\).
\(6x^2 + 26x + 8 = 2(3x^2 + 13x + 4)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(3\times4 = 12\) and add to \(13\). The numbers are \(12\) and \(1\).
\(3x^2 + 13x + 4 = 3x^2 + 12x + x + 4\)
Group the terms:
\(=(3x^2 + 12x) + (x + 4)\)
Factor out the GCF from each group:
\(= 3x(x + 4) + 1(x + 4)\)
Factor out \((x + 4)\):
\(=(3x + 1)(x + 4)\)

Step 3: Combine the factors

Putting it all together, we have:
\(6x^2 + 26x + 8 = 2(3x + 1)(x + 4)\)

Step 1: Factor out the GCF

The GCF of \(3x^2\), \(9xy\), and \(6y^2\) is \(3\).
\(3x^2 + 9xy + 6y^2 = 3(x^2 + 3xy + 2y^2)\)

Step 2: Factor the quadratic

We need two numbers that multiply to \(1\times2 = 2\) and add to \(3\). The numbers are \(2\) and \(1\).
\(x^2 + 3xy + 2y^2 = x^2 + 2xy + xy + 2y^2\)
Group the terms:
\(=(x^2 + 2xy) + (xy + 2y^2)\)
Factor out the GCF from each group:
\(= x(x + 2y) + y(x + 2y)\)
Factor out \((x + 2y)\):
\(=(x + y)(x + 2y)\)

Step 3: Combine the factors

Putting it all together, we have:
\(3x^2 + 9xy + 6y^2 = 3(x + y)(x + 2y)\)

Step 1: Factor the quadratic

We need two numbers that multiply to \(2\times25 = 50\) and add to \(15\). The numbers are \(10\) and \(5\).
\(2x^2 + 15xy + 25y^2 = 2x^2 + 10xy + 5xy + 25y^2\)
Group the terms:
\(=(2x^2 + 10xy) + (5xy + 25y^2)\)
Factor out the GCF from each group:
\(= 2x(x + 5y) + 5y(x + 5y)\)
Factor out \((x + 5y)\):
\(=(2x + 5y)(x + 5y)\)

Answer:

\(2(3x + 1)(x + 4)\)

Problem 10: \(3x^2 + 9xy + 6y^2\)