Q. \[(-3x-4)(x^{2}+3x-5)\]

Answer

Multiply each term (distribute):

\[
(-3x-4)(x^2+3x-5)
= -3x\cdot x^2 -3x\cdot 3x -3x\cdot(-5) -4\cdot x^2 -4\cdot 3x -4\cdot(-5)
\]
\[
= -3x^3 -9x^2 +15x -4x^2 -12x +20
= -3x^3 -13x^2 +3x +20.
\]

Final result: \(\displaystyle -3x^3-13x^2+3x+20.\)

Detailed Explanation

Solution

Write the product to be expanded:
\[(-3x-4)(x^{2}+3x-5)\]
Distribute each term of the first factor across the second factor (use the distributive property):
\[
(-3x-4)(x^{2}+3x-5)
= -3x(x^{2}+3x-5)\;-\;4(x^{2}+3x-5)
\]
Multiply term-by-term for each distributed piece:
\[
-3x(x^{2}+3x-5) = -3x^{3}-9x^{2}+15x
\]

\[
-4(x^{2}+3x-5) = -4x^{2}-12x+20
\]
Add the two results and combine like terms:
\[
(-3x^{3}-9x^{2}+15x)\;+\;(-4x^{2}-12x+20)
= -3x^{3}+(-9x^{2}-4x^{2})+(15x-12x)+20
\]

\[
= -3x^{3}-13x^{2}+3x+20
\]

Final answer: \[-3x^{3}-13x^{2}+3x+20\]

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FAQs

Q1: How do I expand \((-3x-4)(x^2+3x-5)\)?

A1: Distribute each term: \((-3x)(x^2+3x-5)+(-4)(x^2+3x-5)\). Compute: \(-3x^3-9x^2+15x-4x^2-12x+20\). Combine like terms: \(-3x^3-13x^2+3x+20\).

Q2: What property is used to multiply these polynomials?

A2: The distributive property (also called distribution) is used: multiply each term of one factor by every term of the other and then combine like terms.

Q3: Is there a shortcut (like FOIL) for a binomial times a trinomial?

A3: No simple FOIL; use distribution. Multiply each term of the binomial by the entire trinomial (two distributions) then combine like terms.

Q4: How do I combine like terms in the result?

A4: Group terms with the same power: from \(-3x^3-9x^2-4x^2+15x-12x+20\) combine \(x^2\) terms: \(-9x^2-4x^2=-13x^2\) and \(x\) terms: \(15x-12x=3x\).

Q5: Can the expanded polynomial be factored further?

A5: The product is \(-3x^3-13x^2+3x+20\). Since \(x^2+3x-5\) has discriminant \(29\) (not a rational square), it’s irreducible over the rationals; the given factorization is already simplest over \(\mathbb{-Q}\).

Q6: What is the degree and leading coefficient of the expanded polynomial?

A6: Degree is \(3\) (highest power \(x^3\)). Leading coefficient is \(-3\) from the term \(-3x^3\).

Q7: What is the end behavior of the polynomial \( -3x^3-13x^2+3x+20\)?

A7: -As \(x\to\infty\), the polynomial \(\to -\infty\); as \(x\to -\infty\), it \(\to \infty\). (Odd degree with negative leading coefficient.)

Q8: How can I check my expansion is correct?

A8: Plug a convenient value (e.g., \(x=1\)): original gives \((-3-4)(1+3-5)=(-7)(-1)=7\). Expanded gives \(-3-13+3+20=7\). Matching values confirm correctness.

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