1. **Stating the problem:** We are given two quadratic expressions and two factored expressions. We will analyze and verify the factored forms and expressions. 2. **Expression 1: $2x^2 + 5x + 3$** - This is a quadratic expression. - To factor it, we look for two numbers that multiply to $2 \times 3 = 6$ and add to $5$. - These numbers are $2$ and $3$. - Rewrite the middle term: $2x^2 + 2x + 3x + 3$. - Factor by grouping: $$2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1)$$ 3. **Expression 2: $12c^2 + 5c - 5$** - To factor, find two numbers that multiply to $12 \times (-5) = -60$ and add to $5$. - These numbers are $10$ and $-6$. - Rewrite the middle term: $12c^2 + 10c - 6c - 5$. - Factor by grouping: $$2c(6c + 5) - 1(6c + 5) = (2c - 1)(6c + 5)$$ 4. **Expression 3: $(5x + 3)(x - 2)$** - Multiply to verify: $$5x \times x = 5x^2$$ $$5x \times (-2) = -10x$$ $$3 \times x = 3x$$ $$3 \times (-2) = -6$$ - Combine like terms: $$5x^2 - 10x + 3x - 6 = 5x^2 - 7x - 6$$ - This is the expanded form. 5. **Expression 4: $(4mtn)(m - 2n)$** - Multiply: $$4mtn \times m = 4m^2tn$$ $$4mtn \times (-2n) = -8mtn^2$$ - The product is: $$4m^2tn - 8mtn^2$$ **Summary:** - $2x^2 + 5x + 3$ factors as $(2x + 3)(x + 1)$. - $12c^2 + 5c - 5$ factors as $(2c - 1)(6c + 5)$. - $(5x + 3)(x - 2)$ expands to $5x^2 - 7x - 6$. - $(4mtn)(m - 2n)$ expands to $4m^2tn - 8mtn^2$. These steps show factoring and expansion of the given expressions.