Find a fully factorised expression for f(x) = (x+1) + (x+1)^2

We are given the function:

\[
f(x) = (x+1) + (x+1)^2
\]

### Step 1: Set a substitution
Let \( y = x + 1 \), which simplifies the expression to:

\[
f(x) = y + y^2
\]

### Step 2: Factor the new expression
We can factor \( y + y^2 \) by first factoring out the common factor of \( y \):

\[
f(x) = y(1 + y)
\]

### Step 3: Substitute back
Since \( y = x + 1 \), substitute back to get the factorized expression in terms of \( x \):

\[
f(x) = (x+1)((x+1) + 1)
\]

Simplifying further:

\[
f(x) = (x+1)(x+2)
\]

Thus, the fully factorized expression is:

\[
f(x) = (x+1)(x+2)
\]

Would you like any more details or have any questions? Here are 5 related questions to consider:

1. How can we expand the factorized form to verify the result?
2. What is the significance of using substitution in this factorization?
3. How would you factor other quadratic expressions of similar form?
4. Can this factorization technique be applied to cubic polynomials?
5. How does factoring help in solving equations involving polynomials?

**Tip:** Always check your factorized form by expanding it to ensure it matches the original expression.

This tutorial explains how to factorize the expression f(x) = (x+1) + (x+1)^2 using substitution and factoring techniques. Learn step-by-step how to rewrite the function as (x+1)(x+2), suitable for students in Grades 9-10.

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