Factoring the Algebraic Expression

We are asked to find the factors of the expression: \((3x + y)^2 + (3x+y)(x+5y)-20(x+5y)^2\). Let's simplify this expression by using substitution.

Using Substitution for Simplification

To make the expression easier to handle, let's substitute:

• Let \(A = (3x + y)\)
• Let \(B = (x+5y)\)

Substituting these into the original expression gives us:

\(A^2 + AB - 20B^2\)

Factoring the Quadratic Form

The expression \(A^2 + AB - 20B^2\) is a quadratic expression in terms of \(A\) and \(B\). We can factor it similar to how we factor a standard quadratic equation like \(z^2 + z - 20\). We need two numbers that multiply to \(-20\) and add up to \(1\). These numbers are \(5\) and \(-4\).

Therefore, we can factor the expression as:

\((A + 5B)(A - 4B)\)

Substituting Back to Original Terms

Now, let's substitute back \(A = (3x + y)\) and \(B = (x+5y)\) into the factored form:

First Factor: \((A + 5B)\)

Substitute the values of A and B:

\((3x + y) + 5(x+5y)\)

Distribute the \(5\):

\(3x + y + 5x + 25y\)

Combine like terms:

\(8x + 26y\)

We can factor out a \(2\) from this expression:

\(2(4x + 13y)\)

This shows that \((4x + 13y)\) is a factor of the original expression.

Second Factor: \((A - 4B)\)

Substitute the values of A and B:

\((3x + y) - 4(x+5y)\)

Distribute the \(-4\):

\(3x + y - 4x - 20y\)

Combine like terms:

\(-x - 19y\)

We can factor out a \(-1\) from this expression:

\(-(x + 19y)\)

This shows that \((x + 19y)\) is a factor of the original expression.

Conclusion on Factors

The original expression factors completely into \(2(4x + 13y) \times -(x + 19y)\), which simplifies to \(-2(4x + 13y)(x + 19y)\).

Based on our analysis:

• Factor I: \((4x+13y)\) is present in the factored form.
• Factor II: \((x+19y)\) is present in the factored form.

Therefore, both \((4x+13y)\) and \((x+19y)\) are factors of the given algebraic expression.