Algebraic Simplification Explained

The question asks us to simplify the given algebraic expression. Simplifying means performing all the indicated operations (like multiplication and addition) and combining similar terms to get the expression in its most compact form.

The expression to simplify is:

$x(9x - 8) + 7(x^2 - 3) + 18$

Step 1: Apply the Distributive Property

We start by applying the distributive property. This involves multiplying the term outside each set of parentheses by every term inside those parentheses.

• Distribute $x$ into the first parenthesis $(9x - 8)$:

  $x \times (9x - 8) = (x \times 9x) - (x \times 8)$

  $= 9x^2 - 8x$

• Distribute $7$ into the second parenthesis $(x^2 - 3)$:

  $7 \times (x^2 - 3) = (7 \times x^2) - (7 \times 3)$

  $= 7x^2 - 21$

Step 2: Substitute and Rewrite the Expression

Now, we replace the parts we just expanded back into the original expression:

$ (9x^2 - 8x) + (7x^2 - 21) + 18 $

Step 3: Combine Like Terms

The next step is to identify and combine terms that are alike. Like terms have the same variable raised to the same power.

• Combine $x^2$ terms: We have $9x^2$ and $7x^2$.

  $9x^2 + 7x^2 = (9 + 7)x^2 = 16x^2$

• Combine $x$ terms: We only have one term with $x$, which is $-8x$. It remains as it is.
• Combine constant terms: We have $-21$ and $+18$.

  $-21 + 18 = -3$

Step 4: Write the Final Simplified Expression

Finally, we assemble the combined terms to form the simplified expression:

$16x^2 - 8x - 3$

This is the most simplified form of the original expression.