Simplify the expression 5(x − 3)(x² + 4x + 1).

Step-by-step solution

1. 1

   Choose an order that keeps the numbers small. Multiplication is associative, so you may take $(x-3)(x^2+4x+1)$ first and save the factor of $5$ for last. Multiplying the $5$ in at the start would force you to carry $5x$ and $-15$ through every partial product.

2. 2

   Distribute x over the trinomial.

   $x(x^{2}+4x+1)=x^{3}+4x^{2}+x$

3. 3

   Distribute −3 over the trinomial. Watch every sign:

   $-3(x^{2}+4x+1)=-3x^{2}-12x-3$

4. 4

   Add the two partial products.

   $x^{3}+(4x^{2}-3x^{2})+(x-12x)-3=x^{3}+x^{2}-11x-3$

5. 5

   Multiply through by 5. This is the step most often skipped:

   $5\left(x^{3}+x^{2}-11x-3\right)=5x^{3}+5x^{2}-55x-15$

   $\boxed{5x^{3}+5x^{2}-55x-15}$

6. 6

   Check by substitution, and beware the answer options. At $x=1$ the original is $5(-2)(6)=-60$, and the expansion gives $5+5-55-15=-60$. ✓ Note that the commonly printed choice $5x^3+x^2-11x-3$ is the product before the $5$ is distributed, and the other listed choices do not match either — so on a multiple-choice version of this item the honest answer is that none of the options is correct.