Math Snap

STEP 1

1. The expression is a quadratic in terms of $(3x + 1)$.
2. We will use substitution to simplify factoring.
3. After substitution, we will factor the quadratic expression.

STEP 2

1. Substitute $u = 3x + 1$ to simplify the expression.
2. Rewrite the expression in terms of $u$.
3. Factor the quadratic expression.
4. Substitute back to express the factorization in terms of $x$.

STEP 3

Substitute $u = 3x + 1$ into the expression:
$$4(3x + 1)^2 - 5(3x + 1) + 1$$ becomes:
$$4u^2 - 5u + 1$$

STEP 4

Now, we have a quadratic expression in terms of $u$:
$$4u^2 - 5u + 1$$

STEP 5

Factor the quadratic expression $4u^2 - 5u + 1$.
To factor, look for two numbers that multiply to $4 \times 1 = 4$ and add to $-5$. These numbers are $-4$ and $-1$.
Rewrite the middle term using these numbers:
$$4u^2 - 4u - u + 1$$ Group terms:
$$(4u^2 - 4u) - (u - 1)$$ Factor each group:
$$4u(u - 1) - 1(u - 1)$$ Factor out the common factor $(u - 1)$:
$$(4u - 1)(u - 1)$$

Substitute back $u = 3x + 1$ into the factored expression:
$$(4(3x + 1) - 1)((3x + 1) - 1)$$ Simplify each factor:
First factor:
$$4(3x + 1) - 1 = 12x + 4 - 1 = 12x + 3$$ Second factor:
$$(3x + 1) - 1 = 3x$$ Thus, the factored expression is:
$$(12x + 3)(3x)$$