Simplifying Algebraic Expressions Step-by-Step

The problem asks us to simplify the algebraic expression: $x(3x - 5) + 5(x^2 – 3) + 16$

Step 1: Distribute the terms

First, we distribute the $x$ into the first parenthesis $(3x - 5)$ and the $5$ into the second parenthesis $(x^2 – 3)$.

• Distributing $x$: $x \times (3x) = 3x^2$ $x \times (-5) = -5x$ So, $x(3x - 5)$ becomes $3x^2 - 5x$.
• Distributing $5$: $5 \times (x^2) = 5x^2$ $5 \times (-3) = -15$ So, $5(x^2 – 3)$ becomes $5x^2 - 15$.

Step 2: Rewrite the expression

Now, substitute these expanded forms back into the original expression:

$(3x^2 - 5x) + (5x^2 - 15) + 16$

Step 3: Combine like terms

Next, we group and combine terms that have the same variable and exponent.

• Combine the $x^2$ terms: $3x^2 + 5x^2 = 8x^2$
• Combine the $x$ terms: There is only one $x$ term, which is $-5x$.
• Combine the constant terms: $-15 + 16 = 1$

Step 4: Final Simplified Expression

Putting the combined terms together, we get the simplified expression:

$8x^2 - 5x + 1$

This result matches one of the provided options. The process involves applying the distributive property and then combining like terms to simplify the polynomial expression.